QuestionFind the intersection of the complements: $A^{\prime} \cap B^{\prime}$ where $U=\{1,2,3,4,5,6,7\}$ , $A=\{1,2,5,7\}$ , $B=\{1,4,5\}$ .

Studdy Solution

STEP 1

Assumptions1. The universal set U is {1,,3,4,5,6,7} . Set A is {1,,5,7}
3. Set B is {1,4,5}
4. We are looking for the intersection of the complement of A and the complement of B, denoted as $A^{\prime} \cap B^{\prime}$

STEP 2

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

STEP 3

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

STEP 4

Calculate the complement of set A.
$A^{\prime} = \{3,4,6\}$

STEP 5

Next, we need to find the complement of set B, denoted as $B^{\prime}$ . The complement of a set is the set of all elements in the universal set that are not in the given set.
$B^{\prime} = U - B$

STEP 6

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

STEP 7

Calculate the complement of set B.
$B^{\prime} = \{2,3,6,7\}$

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.
$A^{\prime} \cap B^{\prime} = A^{\prime} \cap B^{\prime}$

STEP 9

Plug in the values for $A^{\prime}$ and $B^{\prime}$ to calculate their intersection.
$A^{\prime} \cap B^{\prime} = \{3,4,6\} \cap \{2,3,6,7\}$

STEP 10

Calculate the intersection of $A^{\prime}$ and $B^{\prime}$ .
$A^{\prime} \cap B^{\prime} = \{3,6\}$ So, $A^{\prime} \cap B^{\prime} = \{3,6\}$ .

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