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QuestionFind A′∩B′A^{\prime} \cap B^{\prime} given U={1,2,3,4,5,6,7}U=\{1,2,3,4,5,6,7\}, A={1,3,4,6}A=\{1,3,4,6\}, B={3,5,6}B=\{3,5,6\}.

Studdy Solution

STEP 1

Assumptions1. The universal set U is {1,,3,4,5,6,7}\{1,,3,4,5,6,7\} . Set A is {1,3,4,6}\{1,3,4,6\}
3. Set B is {3,5,6}\{3,5,6\}

STEP 2

First, we need to find the complement of set A, denoted as A′A'. The complement of a set A is the set of all elements in the universal set that are not in A.
A′=U−AA' = U - A

STEP 3

Now, plug in the given values for the universal set U and set A to calculate A′A'.
A′={1,2,3,,5,6,7}−{1,3,,6}A' = \{1,2,3,,5,6,7\} - \{1,3,,6\}

STEP 4

Calculate the complement of set A.
A′={2,,7}A' = \{2,,7\}

STEP 5

Next, we need to find the complement of set B, denoted as B′B'. The complement of a set B is the set of all elements in the universal set that are not in B.
B′=U−BB' = U - B

STEP 6

Now, plug in the given values for the universal set U and set B to calculate B′B'.
B′={1,2,3,4,5,6,}−{3,5,6}B' = \{1,2,3,4,5,6,\} - \{3,5,6\}

STEP 7

Calculate the complement of set B.
B′={1,2,4,7}B' = \{1,2,4,7\}

STEP 8

Now that we have the complements of sets A and B, we can find their intersection, denoted as A′∩B′A' \cap B'. The intersection of two sets is the set of elements that are common to both sets.
A′∩B′=A′∩B′A' \cap B' = A' \cap B'

STEP 9

Plug in the values for A′A' and B′B' to calculate A′∩B′A' \cap B'.
A′∩B′={2,5,7}∩{,2,4,7}A' \cap B' = \{2,5,7\} \cap \{,2,4,7\}

STEP 10

Calculate the intersection of A′A' and B′B'.
A′∩B′={2,7}A' \cap B' = \{2,7\}So, A′∩B′={2,7}A' \cap B' = \{2,7\}.

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