Math  /  Discrete

QuestionA 00 80 18 B 62 62 U 49 Find the number of elements in each of the following sets. Draw your Venn Diagram on paper and show how you find each set. U === 209 BC == ACUB= (AUB) = C (AUB)C= = ACBC=

Studdy Solution

STEP 1

What is this asking? We need to find the number of elements in different combinations of sets A, B, and C, given a Venn diagram with A and B, along with some element counts.
Oh, and there's a typo - it should be B and C, not A and C!
Don't you worry, we'll figure it out! Watch out! It's easy to miscount elements in overlapping sections, so let's be super careful and double-check our work!
Also, remember that CC isn't shown in the diagram, so we'll have to think strategically about how to handle it.

STEP 2

1. Find the elements only in A.
2. Find the elements only in B.
3. Find the union of A and B.
4. Find the complement of the union of A and B.
5. Find the intersection of B and C.
6. Find the intersection of the complements of A and B.

STEP 3

We know A has a **total** of 8080 elements, and 1818 of those are **shared** with B.
To find the elements **exclusively** in A, we **subtract** the shared elements from the total in A.
This is because we want to **exclude** the elements in both A and B.

STEP 4

8018=6280 - 18 = 62 So, there are 6262 elements **only** in A.

STEP 5

Similarly, B has 6262 elements total, and 1818 are shared with A.
We subtract the shared elements to find the number **exclusively** in B.

STEP 6

6218=4462 - 18 = 44 There are 4444 elements only in B.

STEP 7

The **union** of A and B, written ABA \cup B, includes all elements in A, all elements in B, but we only count the shared elements once!

STEP 8

62+18+44=12462 + 18 + 44 = 124 So, ABA \cup B has 124124 elements.

STEP 9

The **complement** of ABA \cup B, written (AB)C(A \cup B)^C, means everything *outside* of ABA \cup B.
We know the **universal set** UU has 209209 elements.

STEP 10

209124=85209 - 124 = 85 Therefore, (AB)C(A \cup B)^C has 8585 elements.

STEP 11

The problem states that BCB \cap C has no elements, meaning B and C share nothing!

STEP 12

BC=0B \cap C = 0

STEP 13

ACBCA^C \cap B^C represents the elements *outside* of both A and B.
This is the same as (AB)C(A \cup B)^C, which we already calculated!

STEP 14

ACBC=(AB)C=85A^C \cap B^C = (A \cup B)^C = 85

STEP 15

AA only has 6262 elements. BB only has 4444 elements. ABA \cup B has 124124 elements. (AB)C(A \cup B)^C has 8585 elements. BCB \cap C has 00 elements. ACBCA^C \cap B^C has 8585 elements.

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