Part A

False. The symbol means “in”, and corn is not in the set of ingredients

Part B

True. Her beauty potion does require a green apple for some reason 🤷‍♀️

Part C

True. The symbol means “not in”, and it is true that her potion does not require coconut.

Part A

Note that 0 is not a positive (or negative) number, so it is not included

Part B

Here we just directly list the elements:

Part C

Remember that sets can not have repeated elements:

Part D

Scandinavia is a region in Europe that contains Denmark (🇩🇰), Norway (🇳🇴) and Sweden (🇸🇪), the order of the flags doesn’t matter:

S= { 🇩🇰, 🇳🇴, 🇸🇪 }

Part A

We will include all the animals that have some form of fur:

Part B

An amphibian is born in water, is cold blooded and has gills. The only match in this list is the frog

Part C

Most of the animals here are obviously not fish, except maybe the octopus? But an octopus doesn’t have a backbone or skeleton. An octopus is technically a type of mollusk.

So therefore, no animals make it into , so is the empty set:

Part A

The clubs and spades are the only black cards in a deck of cards, so:

Part B

Since there are only six cards in the set above:

Part C

There are 26 possible cards here as shown in the Venn diagram below:

Part D

As we can see in the answer to part c, the black royalty cards are a subset of the set of clubs and spades. In fact, it is a proper subset, so:

Part a

True. Every letter in the first set is in the second set but some letters in the second set are not in the first set

Part b

False. There are some floor coverings that are not in the set of . For example timer floors.

Part c

True. Order doesn’t matter, and since every letter in the left set is on the right as well, then the two sets are equal.

Part d

False. Again, these sets are the same, but doesn’t allow for equality, the right set needs to contain elements that are not in the left set.

Part e

True. This one is a little tricky, but you can always say that the empty set is a subset of any other set. We can say this because:

Every element in the empty set is in the set of prime numbers

The fact that their are no elements int he empty set doesn’t make the statement above false

Part f

True. The order of elements in a set does not matter. Since you have the same elements in both sets, then these sets must be equal.

Part A

We should be methodical about this, so let’s list all of the subsets by particular lengths.

Four Element Subsets

There is only one subset with four elements, the original one:

Three Element Subsets

There are four possible subsets with three elements. You can form them by deleting one element, and there are four possible elements to delete:

Two element Subsets

This time we want to let only two animals free, and there are six ways to do that:

One Element Subsets

This time we only want to set one animal free at a time, so there are again, four possible subsets:

Zero Element Subsets

It turns out that the empty set is also a subset of the original set. The mean old dragon may decide to let nobody free after all. so we also have:

Part B

So the total number of subsets are:

Total Subsets: 1 + 4 + 6 + 4 + 1 = 16

Hmm…. it’s a power of 2 🤔 I Wonder if there’s a good reason for that… Can you figure out why? :)

Problems like these ones are best solved by a picture. We know that:

• A is inside of C

• C is inside of B

So that means that A is also inside of C, so:

Part a

Let’s make a few notes to help us draw the diagram:

• 40 People are outside of the set’s and
  

• That means that you and Daniel know 60 of the people at the party between you

• Since you know 50 of them, then Daniel knows 10 that you don’t know

• Since Daniel knows 20 people, then you have 10 friends in common

We can draw this information on the diagram directly:

Part b

Your mutual acquaintances can be shown on the diagram below directly:

Part c

As explained in part a, this means that you both have 10 friends in common:

Part 1

So for the even numbers, this is easy to write down:

A = {2, 4, 6, 8, 10, 12, …}

Now for the numbers where is prime,

• We start by testing the number 1 - but that doesn’t work because , which is not prime

• Then we test the number 2, and we find that , which is in fact prime, so we include it

If we continue this, we can write out the set directly:

B = {2, 4, 6, 8, 10, 12, …}

Finally, for the set C, we just repeat the same process:

• For the number 1, we test if 1 and have a common digit, and they do not, so we exclude it.

• For the number 2, we test if 2 and have a common digit of 2, so we include the number 2

• Similarly, if we tried 4, it has a common digit with , there is a common digit of 4. So we include 4

Continuing this, we get the set:

C = {2, 4, 6, 8, 10, 12, …}

On the surface, these sets seem to be the same.

Part 2

The even numbers just keep increasing by 2

A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …}

Part 3

So for 10, we can find:

And 107 is prime, so 10 is in the set .

Part 4

We can try the next few numbers after 12. We include them if is a prime number.

So therefore:

B = {2, 4, 6, 8, 10, 12, 16, 18, …}

Part 5

So in this case:

• The number n is 12

• 6n is 72

• 12 and 72 both share a 2

• So then 12 is in the set

Part 6

We include values if and have some digits in common

So this means that:

C = {2, 4, 6, 8, 10, 12, 14, 16, 17, 18, 19, …}

Part 7

Since all of these sets contain uncommon elements, then they are not equal at all.