combination method

>Summer Fruit Smoothie<

INGREDIENTS
Bottom layer: (mango)
1 banana
2 frozen mangoes
1 nectarine
1 cup coconut water

Top layer: (raspberry)
1 banana
1 1/2 cups frozen raspberries
1 cup coconut water

METHOD
Combine and blend the ingredients for the mango smoothie. Pour into a jar until it’s half way full.
Combine and blend the ingredients for the raspberry smoothie. Pour into a jar, filling it up to the top. Top with frozen raspberries/anything else you would like.

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Stretch & Tone routines are some of the most popular in the ZGYM. They combine different methods of movement such as yoga, Pilates, bar and more. It’s a gentle way to awaken even the smallest muscles in your body and help with recovery and rejuvenation.
I’ll see you in the ZGYM at ZuzkaLight.com
Outfit from @popflex_active
#zgym #zuzkalight #stretch #tone #yoga #pilates #barmethod #fitness #workoutmotivation #fitnessmotivation #flexibility

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nowimthevillaininyourhistory said:

I usually have no problem with skipping time in my story, but my current story takes place over almost twenty years and whenever I skip a few years I feel like there are so many things that happened in that time that should have been shown, but which I only tell the reader about. Also my characters change a lot during those years because their lives go on, but I don’t want to confuse my readers. So any tips for skipping 2-5 years without making my manuskript too massive and/or boring? Thank you!


It really depends on just how much is actually happening during those skipped years. Sometimes, it’s little enough that you can use a combination of the following methods, woven into the story, to catch the reader up after a time skip:

1) Exposition - while you certainly don’t want a big info dump of what happened during the skipped period, you can certainly use exposition here and there to explain important things. Look for places (within the scenes following the time skip) where these past developments are relevant and put them in there. For example, let’s say one of your characters had a baby during the time skip. If there is a scene where she is playing with her baby, that would be a good time for a little exposition to catch the reader up on that.

2) Flashbacks - flashbacks are kind of the pariah of “catching the reader up” methods, because they are frequently misused and overused. However, there are times that they can be a great way to illustrate something important the reader missed during a time skip. For example, if something really major happened to a character during the time skip, and then something triggers that memory, a flashback could work. Just don’t overuse this method.

3) Dialogue - people tell each other things, so having one character tell another character about things that happened is another great way to cover something that happened during a time skip:

”Where’s Danny?”

Clyde’s face fell. “Oh. You don’t know?”

”Oh God, what?” Tina asked, feeling uneasy.

”His ship was shot down in the outer sector two years ago. His entire crew was lost.”

4) Recollection - if your story is written in first-person, this will probably fall under exposition, except that rather than just telling the reader something, the character is remembering it. This isn’t the same as a flashback. In flashbacks, the memory is so vivid that the person feels like they’re living it all over again. Recollected memories can be somewhat vivid–like remembering sounds and smells–but it doesn’t feel as real. A flashback is, “Suddenly, I am back in the streets of Shanghai, my heart pounding as I run from the men pursuing me.” A memory is, “I am reminded of my days on the Shanghai streets, so often on the run, being chased by those who would turn me in to the authorities.” One is reliving, one is recalling.

If you can use a combination of these methods to weave the lost details into the story, you’re probably fine.

However, if so much happens during the skipped years that weaving the lost details in this manner would overwhelm what’s happening now–then you need to consider that skipping these events may not be in your story’s best interest. You should never skip important events just to make your story shorter. If something is important enough to be covered, cover it. If your story is going to be really long, break it into multiple stories. In other words, if you’re writing a novel, write a series. :)

Honestly, if so much happens during those years, you probably shouldn’t be skipping them. Skipping important scenes to make the story shorter isn’t going to work. Instead, I think you have to consider the fact that you may have to turn your story into a series instead of trying to skip these big chunks of time. Because, the reality is, you can’t spend a lot of time after every time skip going back over all the important things the reader missed. But you also don’t want to flood the story with flashbacks and recalled memories.

———————————————————————
Have a writing question? I’d love to hear from you! Please be sure to read my ask rules and master list first or your question will not be answered. :)

The Best Vegan Mac and Cheese (v/gf/nut-free/soy-free)

That’s right - a vegan, gluten-free, nut-free, soy-free mac and cheese that is the absolute best that I’ve ever had. I made this for Christmas Eve, and my whole extended, non-vegan family loved it. This is a biiiiiig batch that makes about 2 party sized trays, so definitely half this recipe if you are making it for a dinner-casserole size. I was going to do this recipe again in a smaller batch and post it another time, but I just had to share it with you all. I hope you really give it a try! I’ve done mac and cheese in the past with a roux and daiya, and I’ve seen it before done with potatoes, carrots, and cashews… So I thought I would combine the roux method and with potatoes and carrots to make this incredibly creamy, deceptively vegan, mac and cheese. Next time I do this, I’m going to find a way to not use onion powder and garlic to make it fodmap-friendly. 

Remember, this makes two large casseroles - cut this recipe in half if you are making it for an average dinner

Ingredients:
- 1 900g bag of macaroni (use gf if needed)
- 4 tbsp olive oil
- 4 tbsp All-Purpose flour (use gf if needed)
- 1 946ml box of original rice milk (or use any unsweetened plant-based milk)
- 1 cup of diced carrots (boiled til soft)
- 2 ½ cups of diced white or yellow potatoes (peeled, and boiled til soft)
- 2 cloves finely minced garlic
- ½ tbsp onion powder
- 1 cup nutritional yeast
- salt and pepper to taste (maybe ½ tbsp each)
- bread crumbs and vegan cheese shreds for baking (optional)

Instructions:
1) Prepare your noodles according to the package instructions, set aside until needed
2) In a large sauce pan on medium heat, add oil and flour and whisk until smooth. Add half of your rice milk. Increase heat to medium high and continue to whisk until smooth. Add the garlic, and remove from heat.
3) In a food processor or blender, blend the carrots and potatoes with some rice milk until smooth. You don’t have to be too precise with the rice milk because it will all end up going in the sauce.
4) Put add the blended veggies into the sauce pan, and return to medium heat. Add salt, pepper, onion powder and nutritional yeast. Also add any remaining rice milk. Continue cooking until fully warm. If it is looking too thick, you can add about ¼-½ cup of water to thin. At this point, you can add in your noodles and mix. Serve warm.
5) If you wish to bake your mac and cheese, add the pasta to a lightly greased casserole dish, sprinkle with bread crumbs and vegan cheese (if using), and bake at 400 degrees for about 20 minutes, until the edges are brown and the bread crumbs are toasted. 
6) ENJOY!!

>Raw Vegan Banana Caramel Chocolate Nicecream Cake<

INGREDIENTS
Base:
10 dates soaked and mashed
1 ½ cups oats
2 tbsp maple syrup
1 tbsp desiccated coconut
Pinch of cinnamon

1st layer:
3 frozen bananas

2nd layer:
3 frozen bananas
10 dates soaked and mashed
4-5 drops vanilla essence
1 tsp maca powder

Topping:
1 frozen banana
2 tbsp cacao powder
1 tsp acai powder
1 tbsp maple syrup

METHOD:
Combine ingredients for base, set into a tray. Combine ingredients for first layer, blend until smooth, pour over base. Repeat with the ingredients for the second layer and set in the freezer. When it hardens, combine ingredients for the topping, blend and set in the freezer until the topping is hard. Top with crushed nuts and Buckinis.

I’ve gotten an overwhelming response to my classpect/moon post, but amidst all the answers, there have been a surprising number who don’t know what their Homestuck moon is, or how to find out! I decided to gather my favorite methods/observations into one convenient post. There’s no one correct way to place somebody, and I actually use different combinations of these methods, but they should shed a little light on the subject. In the end, you’ll want to stick with the moon you feel is right for you or the person/character you’re sorting! If enough people show interest, I may also do a masterpost of classpect tests/methods, but for now, I hope this helps somebody out.

(Unfortunately, since some of these have been passed around and repeated so many times in the fandom, it’s become difficult to find the original source of the idea. If you have a source, please let me know.)

  1. Prospit dreamers are Heroes who need to learn to be Warriors, and Derse dreamers are Warriors who need to learn to be Heroes. Alright, so this is pretty abstract. Still, I like it because it puts into words something I’ve always noticed I’ve been unable to articulate, so, y’know, to each their own.
  2. Prospit dreamers keep their beds neat or their recuperacoons visible in their main living space. Derse dreamers have messy beds or keep their recuperacoons separate from their living area, out of sight. This doesn’t really apply outside of canon, since a messy bed can be attributed to lots of unrelated factors (and I’d hope ya’ll don’t have actual recuperacoons), but it’s definitely something to keep in mind if you’re designing/sorting a character. Vriska and Terezi are the exceptions to this rule, as Prospit dreamers whose recuperacoons are not shown in their rooms. I’m not sure why this is, but it seems significant that it’s those two in particular.
  3. Prospit dreamers usually have passive masculine-tendency or active feminine-tendency god tier classes. Derse dreamers usually have active masculine or passive feminine classes. This one is my favorite, because it’s mostly concrete but still allows for people to be people and for sessions to have irregularities, as they generally do. If you don’t know what categories your class falls under, there’s a chart below that should help, along with a handy-dandy flowchart that covers pretty much everything you need to know about this option. Exceptions: Any troll that shares a class with a human is on the opposite moon, except for Tavros, Nepeta, and Eridan (who happen to be class buddies with Jake, Roxy, and Dirk, respectively. Not sure if this means anything). Kanaya and Calliope dream on Prospit, despite their passive classes, possibly because they are Space players. Aradia and Damara, both active gals, dream on Derse, but it’s difficult to say whether this is because they share classes with kids, because they are Time players, or both. There is little evidence for aspects affecting dream moons, except for Space and Time (which are consistent throughout the comic), so I wouldn’t worry about it unless you fall under either of those.

Keep in mind that Sburb/Sgrub sessions are shown to have an even number of Prospit and Derse dreamers. You may want to take this into account in fan sessions.

Class chart:

Heir, Mage, Seer, Rogue, and Knight are shown to be more on the neutral side, but in the interest of simplicity, they’ve been assigned genders based on several elements found in canon (except for Mage, that one might be the only True Gender Neutral Class™, but I’ve assigned an M to it to keep things symmetrical). Assuming the Alpha trolls dream on the same moons as their counterparts, Kankri, Rufioh, and Latula all identify as the opposite binary gender from that to which their class is traditionally assigned, but they dream on the moon that fits with their gender preferences (ex. Kankri is passive masculine and thus dreams on Prospit, despite Seer being traditionally feminine) Ultimately, the gender is up to you.

Tumblr screwed up the flowchart, but here’s a link to a full sized version.

None of these methods decide anything for certain, even the more concrete ones. Everything is ultimately up to you. Go forth, Homestucks, and pleasant dreams!

(Also, it’s interesting to note that Sollux has his recuperacoon visible in his room, indicative of a Prospit dreamer, but method 1 points to Derse.¯\_(ツ)_/¯)

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I think the killer legs series is one of my favourites in the ZGYM. I just like lower body workouts and making my legs strong. These routines are super effective for those of you who’d like to see more shape and definition in your legs from calves to glutes. We’re combining bar method with cardio and strength endurance circuit training. And the best part is that you can do these workouts at home. You can try to do 10 reps for each exercise you see in this video (on each leg if it’s a unilateral exercise) and if you want to do all the exercises in this workout with me, then I’ll see you in the ZGYM at ZuzkaLight.com
#legs #fatburn #weightloss #lean #exercise #fitness #workout #video #workoutmotivation #zuzkalight #zgym

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2

Pain, hurts, but I won’t cry! Parade? Marade? Just clap a little more!

Aaand here’s the hardest two to plan for in this series because the ‘combine outfits’ method I’m using doesn’t exactly work with monsters eAe;; But yeah Ivan and Myléne are the only two wearing the iconic Matryoshka-style hoodies out of the whole class! With this I’m almost halfway there~ contemplating whether I want to finish DJWifi or Sabrina and Chloe next, they’re both fun.

MIRACULOSHKA series: [x] [x]

alright so the other day someone sent me an ask about fandom type sigils (like with fandom alphabets like tolkein elvish and Galifreyan from doctor who), which sounds like an amazing idea. however I don’t know enough about elvish, and I personally find Galifreyan slightly to difficult to write in for my own personal tastes. However, I do love the bionicle alphabet, and thought it might be cool to do sigils in this alphabet. 


Which is tricky because usually I make sigils with the cross out the vowels and repeated letters and make them into a shape then charge them method. So how would I go about doing that with the bionicle alphabet which is all circular in terms of letter shape. 

This was my solution:

typically bionicle language is written out the same as english with the letters replaced. And so my method for combining them into a sigil involves concentric circles as a means of combining the letters. (note I didn’t charge this sigil)


Anyway I really like this style of sigil craft, and perhaps will use it more in the future.

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The Killer Legs workout series in the ZGYM is all about conditioning your legs and building the muscles of your calves, thighs and glutes. We’re combining the bar method with weight training and high intensity cardio. Fun & results is what always matters to me 😜 I’ll see you guys tomorrow in the ZGYM and we’ll do this workout together!
#zuzkalight #zgym #fitness #training #legs #calves #thighs #cardio

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Freckles Tutorial:

For all your cosplay/non cosplay make-up needs:

Alrighty so I usually combine a few methods of freckle making when I do freckles for cosplays such as Marco Bodt.

(This is kinda long so I’m gonna put it after the cut)


Materials:

-Eyeliner Pencils (I use 3 shades of brown, got them for 99 cents each at Fred     Meyers)

-A large make-up brush

-Some sort of bronzer/darker loose powder

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Emeralds and oxygen: Back to the source.

Finding out where emeralds come from is a tricky business. Laboratory gemmologists asked to write a coloured stone origin report and archaeologists wishing to trace stones in ancient jewellery need this knowledge, and a variety of techniques are used in combination in order to arrive at a robust conclusion.

Keep reading

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Our Jiu Jitsu classes in Portland shape people with foundations. Here’s a small clip of one of our brown belt instructors demonstrating the standard hip escape. Coach Erik then combines the method with the invisible force of “yielding” which is how he gets out so fast from his partners mount. In combination with another basic movement, taking the back from guard, it becomes s vey powerful setup to quickly get the best position on the ground.

The best part of this clip is we caught a white belt and blue belt performing the movements as they worked it out in class. That’s how learning BJJ develops at our class at RCW in Tigard. It’s a step by step progression using forces like yielding and leverage so anyone big, tall, or small can learn real Gracie Jiu Jitsu.

Head of the Buddha

  • Place of origin: Afghanistan (probably, made)
  • Date: 4th century - 5th century
  • Materials and technique: Stucco with polychrome

This head of the Buddha, once part of a large-scale narrative painted in high relief, is a good example of the Gandharan art style, which was born and thrived in the Greek-ruled states created in Central Asia following Alexander’s conquests. The art style combined Hellenistic methods to represent motifs local to Central Asia, as it can be seen especially in the hair-knot (ushnisha), treated in a Greek way.

A spell for Chronic Pain Relief

This spell has been constructed for hazelplanchette​.

This spell uses a combination of methods to boost the desired result! I have kept it decently simple, as you have chronic pain so i would imagine that you wouldn’t like to spend a long time doing spells. The theme of this spell is a potent and fiery one. What the intentions are is to no longer allow the pain to control you, and let it know it has no business with you any longer.

You will need:

-2 Tea-light candles (any colour) also, a lighter or matches, heatproof surface, and safe candle holder! 

-A heatproof bowl of some sort.

-A crystal you associate with pain relief (some examples are: blue lace agate, amethyst, rose quartz, earthy green stones)

-A sigil for pain relief: I sent you one i have designed for you! Also, a non-toxic pen or marker, a sharp object to inscribe the candles and also a piece of durable paper with the sigil written on it in a strong and bold colour.

-Tarot cards: Strength and The Tower

The Spell:

First of all, have the candles inscribed with the sigil, the crystals charged and the sigil written on the paper.

Set up your work area, if you like music, put some on. You will want something with a fire/earthy vibe and a warrior perception. So rough and soulful music will work, as well as something with tribal drums or anything that gets your blood pumping! Have the candles in front of you, the tarot cards in the middle, and the crystals around them. Place the sigil in your lap.

Take a deep breath and get your mind, body and soul aligned and in the right mood for spell-casting! Light the two candles and listen to that music. Feel the heat of your body rising along with your blood pressure, feel your heart beating and your blood pumping.

Now, imagine you pain to be a black slime ball, or maybe a garden that is filled with poisonous plants that are doing your body harm. Get that energy that you feel within you and really let it peak. Do what you can to push the two energies (the black painful slime and your fiery heartbeat) into the sigil. Just imagine them being locked in there together, fighting one another. Know in your heart that your personal strength will win! Scrunch this paper up into a ball and place it on top of the tarot cards. Strength is there to represent your life force and your strength of heart, mind and soul, and the Tower represents the way the pain makes your life crash out of control. If you can, burn the paper and know that the black slime of pain is burning into oblivion and your strength is like a phoenix, rising through the ashes! 

Collect as many of the ashes as you can in a jar or something, and rub a tiny dot of them onto the place of pain, or somewhere close by. This will imbue you with the strength of your inner soul. You can overcome this! Whenever the pain returns, draw the sigil on your skin with a makeup pencil or something safe and smudge a dot of the ashes on top. See the black slime pain being overthrown by your strength. 

The more you use the sigil, the more power and energy it will retain and the quicker it will work each time. It is your unique symbol that represents strength and overcoming the pain you experience. If you need more ashes, or feel that the spell needs a real Boost, feel free to do this again! 

Also you can change any aspect of the spell that you wish! 

I hope you like it and you find relief soon!

>Raw Vegan Raspberry Nicecream Bars<

INGREDIENTS
Base:
10 dates soaked
1 cup oats
2 tbsp maple syrup
1 tbsp dissected coconut
1 tsp coconut oil
Pinch of cinnamon

Filling:
1 frozen banana 1 fresh
1/3 cup raspberries thawed/fresh
2-3 drops vanilla essence

Raspberry layer:
2 cups frozen raspberries
Splash of coconut water

Topping:
1 ½ tbsp coconut oil melted
1 ½ tbsp cacao powder
2 tbsp maple syrup

METHOD:
Combine ingredients for the base, press into a tray and set in the freezer.
Combine ingredients for the filling and blend until creamy texture. Layer over base and set in the freezer for an hour.
After the filling hardens, blend 2 cups of frozen raspberries and layer over the filling. Leave in the freezer for an hour.
Combine coconut oil, cacao powder and maple syrup until a sauce forms and pour over the raspberry layer. While it’s melted top with raspberries and set in the freezer until the chocolate top hardens.
When the topping is hard, top with blackstrap molasses and more berries if you like. Enjoy! x

Instagram: @annietarasova

Set Operations and the Laws of Set Theory

Binary Operations
After learning how to count, a student usually uses methods for combining counting numbers. First, this is accomplished through addition. Usually the student uses the universe of positive integers ℤ⁺, where the addition of two elements from ℤ⁺ result in a third element of ℤ⁺ called the sum. This is also true for the operation of multiplication. Therefore, only results of elements from ℤ⁺ would be produced and no other element from other universes will appear.

Therefore, the addition and multiplication operations of positive integers are closed binary operations on ℤ⁺. For example, when a + b, where a,b ∈ ℤ⁺, is computed, there are two operands a,b. This is why it is called binary operations. Since a + b ∈ ℤ⁺ when a,b ∈ ℤ⁺, then the binary operation of addition on ℤ⁺ is closed. However, the binary operation of nonzero division on ℤ⁺ is not closed. For example, 1,2 ∈ ℤ⁺, but ½ ∉ ℤ⁺. The binary operations of nonzero division is closed on ℚ⁺.

Binary Operations for Sets
For the sets A,B ⊆ 𝓤, the following are closed binary operations for sets A,B on 𝓤:

1. Union of A and B: A ⋃ B = {x | x ∈ A ᴠ x ∈ B}
2. Intersection of A and B: A ⋂ B = {x | x ∈ A ᴧ x ∈ B}
3. Symmetric Difference of A and B: A ∆ B = {x | (x ∈ A ᴠ x ∈ B) ᴧ x ∉ A ⋂ B}

Note that the symmetric difference of sets A and B is similar to the logical connective exclusive “or” ⊻.

Since A,B ⊆ 𝓤, then A ⋃ B, A ⋂ B, A ∆ B ⊆ 𝓤. Therefore ⋃, ⋂, and ∆ are closed binary operations on ℘(𝓤), and so ℘(𝓤) is closed under these binary operations.

Example
With 𝓤 = {1,2,3,…,9,10}, A = {1,2,3,4,5}, B = {3,4,5,6,7}, and C = {7,8,9}, then:
A ⋂ B = {3,4,5}
A ⋃ B = {1,2,3,4,5,6,7}
B ⋂ C = {7}
A ⋂ C = ∅
A ∆ B = {1,2,6,7}
A ⋃ C = {1,2,3,4,5,7,8,9}
A ∆ C = {1,2,3,4,5,7,8,9}

Conjunctive Simplification and Disjunctive Amplification for Sets
If A,B ⊆ 𝓤, then by the Rule of Conjunctive Simplification:

x ∈ A ⋂ B
=> x ∈ A ᴧ x ∈ B
=> x ∈ A

By the Rule of Disjunctive Amplification:

x ∈ A
=> x ∈ A ᴠ x ∈ B
=> x ∈ A ⋃ B

Disjoint Sets
Let the sets S,T ⊆ 𝓤. Then the sets S and T are disjoint, or mutually disjoint, when S ⋂ T = ∅.

Therefore, if S and T are disjoint, then S ∆ T = S ⋃ T, and for any sets A,B ⊆ 𝓤, then A ∆ B ⊆ A ⋃ B.

Subtraction and Negation
The set of positive integers ℤ⁺ contains 2 and 5, however 2 – 5 = -3 ∉ ℤ⁺. Therefore, the binary operation of subtraction is not closed for ℤ⁺, but it is closed on the set of all integers ℤ.

Since the set of integers ℤ contains negatives, the unary, or monary, operation of negation, where the negative of a number is taken, is closed on the set ℤ.

Complement Sets and Relative Complement Sets
For a set A ⊆ 𝓤, the complement of A, denoted as Ᾱ, 𝓤 – A, or A’, is the set of all x in 𝓤 such that x is not in set A. Therefore, Ᾱ = {x | x ∈ 𝓤 ᴧ x ∉ A}.

For every universe 𝓤 and every set A ⊆ 𝓤, it is found that Ᾱ ⊆ 𝓤. Therefore, ℘(𝓤) is closed under the unary operation defined by the complement.

For A,B ⊆ 𝓤, the relative complement of A in B, or the difference of A and B, denoted as B – A, or B ⧹ A, is the set of all x in set B such that x is not in set A. Therefore, B – A = {x | x ∈ B ᴧ x ∉ A}.

Example
With 𝓤 = {1,2,3,…,9,10}, A = {1,2,3,4,5}, B = {3,4,5,6,7}, and C = {7,8,9}, then:
B – A = {6,7}
A – B = {1,2}
A – C = A
C – A = C
A – A = ∅
𝓤 – A = Ᾱ

Example
For 𝓤 = ℝ, let A = [1,2] and B = [1,3). Then:
A = {x | 1 ≤ x ≤ 2} ⊆ {x | 1 ≤ x < 3} = B
A ⋃ B = {x | 1 ≤ x < 3} = B
A ⋂ B = {x | 1 ≤ x ≤ 2} = A
B̄ = (-∞,1) ⋃ [3,+∞) ⊆ (-∞,1) ⋃ (2,+∞) = Ᾱ

Relations Between Binary Operations and Complements
For any universe 𝓤 and any sets A,B ⊆ 𝓤, the following statements are equivalent:

1. A ⊆ B
2. A ⋃ B = B
3. A ⋂ B = A
4. B̄ ⊆ Ᾱ

The Laws of Set Theory
For any sets A,B,C from a universe 𝓤:

1. Law of Double Complement

2. DeMorgan’s Laws

3. Commutative Laws

4. Associative Laws

5. Distributive Laws

6. Idempotent Laws

7. Identity Laws

8. Domination Laws

9. Inverse Laws

10. Absorption Laws

Principle of Duality and Sets
As with the laws from the second law to the tenth law of logic, the same laws for sets also have paired statements that are duals of each other, where one statement from the pair is obtained from the other by either replacing the union ⋃ with the intersection ⋂, or replacing the empty set ∅ with the universe 𝓤, and vice versa.

Let s be a general statement containing the equality of two set expressions. Each expression may contain one or more occurrences of sets, such as A, Ᾱ, B, B̄, etc., one or more occurrences of the empty set ∅ or the universe 𝓤, and only the set operations union ⋃ and intersection ⋂. Then the dual of s, denoted as sd, is obtained from s by replacing each occurrence of the empty set ∅ with the universe 𝓤, and vice versa, and replacing each occurrence of the union ⋃ with the intersection ⋂, and vice versa.

Note that the principle of duality can only be applied to theorems.

Duality and Subsets
Let A,B ⊆ 𝓤 and A ⊆ B. Since A ⊆ B <=> A ⋃ B = B, the dual of A ⋃ B = B is A ⋂ B = B. This is logically equivalent to B ⊆ A.

Therefore, the dual of A ⊆ B is B ⊆ A.

Venn Diagrams
When the relations that may exist among the sets that are involved in a set-equality or subset statement is considered, the situation can be investigated graphically using Venn diagrams.

A Venn diagram is constructed where the universe 𝓤 is depicted as the interior of a rectangle, while the subsets of 𝓤 are represented in the interior as circles, or, if there are more than three subsets, closed shapes.

The following Venn diagram demonstrates A and Ᾱ, where A is the shaded region:

The following Venn diagram demonstrates A ⋃ B, which is the shaded region:

The following Venn diagram demonstrates A ⋂ B, which is the shaded region:

The following Venn diagram demonstrates B ⧹ A, which is the shaded region:

The following four Venn diagrams demonstrate one of DeMorgan’s Laws:

(A ⋂ B)’:

Ᾱ:

B̄:

Ᾱ ⋃ B̄:

Membership Tables
Besides Venn diagrams, membership tables are created to establish set equalities.

For any sets A,B ⊆ 𝓤, an element x ∈ 𝓤 has the following four possibilities:

1. x ∈ A, x ∈ B
2. x ∉ A, x ∉ B
3. x ∈ A, x ∉ B
4. x ∉ A, x ∈ B

When the element x is a member of a given set, the number “1” is used to denote this membership. Otherwise, it is denoted as “0”.

Therefore, the following membership table is created for A, B, A ⋃ B, A ⋂ B, and Ᾱ:

The following demonstrates that membership tables help determine if two separate subset statements are logically equivalent:

Since the fourth and eighth column have the same truth value assignments, it can be concluded that the Distributive Law holds.

A Venn diagram is a graphical representation of the membership table. Although the use of Venn Diagrams and membership tables is appealing, neither one of these techniques specify the logic and reasoning that is displayed with element arguments.

Simplifying Set Expressions
The Laws of Set Theory can help simplify set expressions just as the Laws of Logic are used to simplify logical statements.

Example
To simplify the following expression:

Example
To express the following using complement and intersection:

PDF reference: 167

Page 146. Question 1.

For a) (A ⋃ B) ⋂ C = {1,2,3,4,5,8} ⋂ {1,2,3,5,7} = {1,2,3,5}

For b) A ⋃ (B ⋂ C) = {1,2,3,4,5} ⋃ {1,2} = {1,2,3,4,5} = A

For c) C’ ⋃ D’ = {1,3,4,5,6,7,8,9,10} = 𝓤 – {2}

For d) (C ⋂ D)’ = C’ ⋃ D’ = {1,3,4,5,6,7,8,9,10} = 𝓤 – {2}

For e) (A ⋃ B) ⧹ C = {4,8}

For f) A ⋃ (B ⧹ C) = A ⋃ {4,8} = {1,2,3,4,5,8}

For g) (B ⧹ C) ⧹ D = {4,8} \ D = ∅

For h) B ⧹ (c ⧹ D) = B ⧹ {1,3,5,7} = {2,4,8}

For i) (A ⋃ B) ⧹ (C ⋂ D) = {1,2,3,4,5,8} ⧹ {2} = {1,3,4,5,8}

Page 146. Question 3.

a)

Both A and B have 4 and 9, so:

A = {4,9,
B = {4,9,

A ⧹ B states that the elements in the set {1,3,7,11} are in A but not in B, so:

A = {4,9,1,3,7,11}

B ⧹ A states that the elements in the set {2,6,8} are in B but not in A, so:

B = {4,9,2,6,8}

Page 147. Question 5.

a) true
b) true
c) true
d) false, |ℝ⁺| > |ℚ|
e) true
f) true
g) true
h) false, the set ℂ ⋃ ℝ will contain complex numbers
i) false, ℚ* does not contain zero, and so ℚ* ⋂ ℤ cannot be a set with zero

Page 147. Question 7.

a)

Let 𝓤 = {1,2,3}, A = {1}, B = {2}, and C = {3}.
Then A ⋂ C = B ⋂ C = ∅, yet A ≠ B. Therefore, the statement is false.

Page 147. Question 11.

b) A = A ⋃ (A ⋂ B)

Page 147. Question 13.

a)

Let 𝓤 = {1,2,3}, A = {1} and B = {2}. Then {1,2} ∈ ℘(A ⋃ B), however, {1,2} ∉ ℘(A) ⋃ ℘(B). Therefore, the statement is false.

Page 147. Question 17.
a) A ⋂ (B ⧹ A) = A ⋂ (B ⋂ Ᾱ) = A ⋂ Ᾱ ⋂ B = ∅ ⋂ B = ∅.

>Raspberry Choc Nicecream<

INGREDIENTS
Bottom layer: (vanilla)
2 frozen bananas
2-3 drops vanilla essence

Top layer: (choc)
2 frozen bananas
2 tsp cacao powder
1 tbsp maple syrup

+ 1 1/2 cups frozen raspberries

METHOD:
Combine and blend all the ingredients for the vanilla nicecream. Use a food processor preferably. When the texture is smooth, put on the bottom of the jar. Layer with a cup of frozen raspberries.
Combine and blend the ingredients for the choc nicecream. Put on top of the frozen raspberries. Top the nicecream with left over berries. Enjoy! X