Math / Discrete

QuestionUse the Venn diagram below to find the number of elements in the region. 4) $n((A \cup B) \cap C)$ A) 14 B) 33 C) 15 D) 11

Studdy Solution

STEP 1

What is this asking? How many things are in both C and either A or B? Watch out! Don't forget about the elements that are in all three sets!
They count too!

STEP 2

1. Find the union of A and B.
2. Find the intersection of the union with C.

STEP 3

Let's first figure out what $A \cup B$ represents.
This means anything in A, or B, or both!
Imagine it like this: if something is in A, it's in the union.
If it's in B, it's in the union.
If it's in both, it's still in the union just once!

STEP 4

Looking at our Venn diagram, the number of elements only in A is $4$ .
The number only in B is $7$ .
The number in both A and B is $3$ .
So, to find the total number of elements in $A \cup B$ , we add those up: $4 + 7 + 3 = 14$ .
Don't forget the $8$ elements that are in A, B, and C!
Since those $8$ elements are in both A and B, they are also part of the union of A and B.
So we add $8$ to our running total: $14 + 8 = \textbf{22}$ .
There are $\textbf{22}$ elements in $A \cup B$ .

STEP 5

Now, we want to find $(A \cup B) \cap C$ .
This means we're looking for elements that are in both the union we just found ( $A \cup B$ ) and also in C.

STEP 6

We already know that $8$ elements are in A, B, and C.
These are definitely in both $(A \cup B)$ and $C$ .
Also, there are $7$ elements that are only in both B and C, and $2$ elements only in both A and C.
Since these elements are in C, and also in either A or B, they are also part of the intersection we're looking for.

STEP 7

Let's add those up! $8 + 7 + 2 = \textbf{17}$ .
So, there are $\textbf{17}$ elements in $(A \cup B) \cap C$ .

STEP 8

The number of elements in $(A \cup B) \cap C$ is $\textbf{17}$ .
However, this isn't one of the answer choices!
Looking back at the problem, we see that the intersection of A and C is $2$ , the intersection of B and C is $7$ , and the intersection of all three is $8$ .
Adding these up, we get $2 + 7 + 8 = \textbf{17}$ .
It seems there was a typo in the answer choices, and the correct answer isn't listed!

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QuestionUse the Venn diagram below to find the number of elements in the region. 4) $n((A \cup B) \cap C)$ A) 14 B) 33 C) 15 D) 11

Studdy Solution

STEP 1

What is this asking? How many things are in both C and either A or B? Watch out! Don't forget about the elements that are in all three sets!
They count too!

STEP 2

1. Find the union of A and B.
2. Find the intersection of the union with C.

STEP 3

Let's first figure out what $A \cup B$ represents.
This means anything in A, or B, or both!
Imagine it like this: if something is in A, it's in the union.
If it's in B, it's in the union.
If it's in both, it's still in the union just once!

STEP 4

STEP 5

Now, we want to find $(A \cup B) \cap C$ .
This means we're looking for elements that are in both the union we just found ( $A \cup B$ ) and also in C.

STEP 6

STEP 7

Let's add those up! $8 + 7 + 2 = \textbf{17}$ .
So, there are $\textbf{17}$ elements in $(A \cup B) \cap C$ .

STEP 8

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QuestionUse the Venn diagram below to find the number of elements in the region. 4) n((A∪B)∩C)n((A \cup B) \cap C)n((A∪B)∩C) A) 14 B) 33 C) 15 D) 11

Studdy Solution

STEP 1

What is this asking? How many things are in *both* C *and* either A or B? Watch out! Don't forget about the elements that are in all *three* sets!They count too!

STEP 2

1. Find the union of A and B.2. Find the intersection of the union with C.

STEP 3

Let's **first** figure out what A∪BA \cup BA∪B represents.This means anything in A, or B, or *both*!Imagine it like this: if something is in A, it's in the union.If it's in B, it's in the union.If it's in *both*, it's *still* in the union just once!

STEP 4

STEP 5

Now, we want to find (A∪B)∩C(A \cup B) \cap C(A∪B)∩C.This means we're looking for elements that are in *both* the union we just found (A∪BA \cup BA∪B) *and* also in C.

STEP 6

STEP 7

Let's **add** those up! 8+7+2=178 + 7 + 2 = \textbf{17}8+7+2=17.So, there are 17\textbf{17}17 elements in (A∪B)∩C(A \cup B) \cap C(A∪B)∩C.

STEP 8

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QuestionUse the Venn diagram below to find the number of elements in the region. 4) $n((A \cup B) \cap C)$ A) 14 B) 33 C) 15 D) 11

What is this asking? How many things are in both C and either A or B? Watch out! Don't forget about the elements that are in all three sets!
They count too!

1. Find the union of A and B.
2. Find the intersection of the union with C.

Let's first figure out what $A \cup B$ represents.
This means anything in A, or B, or both!
Imagine it like this: if something is in A, it's in the union.
If it's in B, it's in the union.
If it's in both, it's still in the union just once!

Now, we want to find $(A \cup B) \cap C$ .
This means we're looking for elements that are in both the union we just found ( $A \cup B$ ) and also in C.

Let's add those up! $8 + 7 + 2 = \textbf{17}$ .
So, there are $\textbf{17}$ elements in $(A \cup B) \cap C$ .