Math

QuestionFind the intersection of the complements: ABA^{\prime} \cap B^{\prime} where 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} . Set A is {1,3,4,6}
3. Set B is {3,5,6}
4. We are asked to find the intersection of the complements of sets A and B, denoted as ABA^{\prime} \cap B^{\prime}

STEP 2

First, we need to find the complement of set A, denoted as AA^{\prime}. The complement of a set is the set of all elements in the universal set that are not in the set.A=UAA^{\prime} = U - A

STEP 3

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

STEP 4

Calculate the complement of set A.
A={2,,7}A^{\prime} = \{2,,7\}

STEP 5

Next, we need to find the complement of set B, denoted as BB^{\prime}.B=UBB^{\prime} = U - B

STEP 6

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

STEP 7

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

STEP 8

Now that we have the complements of sets A and B, we can find their intersection. The intersection of two sets is the set of all elements that are common to both sets.
AB=ABA^{\prime} \cap B^{\prime} = A^{\prime} \cap B^{\prime}

STEP 9

Plug in the values for AA^{\prime} and BB^{\prime} to calculate the intersection.
AB={2,5,7}{,2,4,7}A^{\prime} \cap B^{\prime} = \{2,5,7\} \cap \{,2,4,7\}

STEP 10

Calculate the intersection of AA^{\prime} and BB^{\prime}.
AB={2,7}A^{\prime} \cap B^{\prime} = \{2,7\}So, the intersection of the complements of sets A and B is {2,7}.

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