Math

QuestionFind the complement of the intersection of sets A and C. Given U={1,2,3,4,5,6,7,8}U=\{1,2,3,4,5,6,7,8\}, A={1,2,3,4}A=\{1,2,3,4\}, C={1,2,3,6,8}C=\{1,2,3,6,8\}.

Studdy Solution

STEP 1

Assumptions1. The universal set U is {1,,3,4,5,6,7,8} . The set A is {1,,3,4}
3. The set C is {1,,3,6,8}
4. We need to find the set (AC)(A \cap C)^{\prime}, which is the complement of the intersection of A and C.

STEP 2

First, we need to find the intersection of sets A and C. The intersection of two sets is the set of elements that are common to both sets.
AC={xxAandxC}A \cap C = \{x x \in A \, and \, x \in C\}

STEP 3

Now, plug in the given values for sets A and C to find their intersection.
AC={1,2,3,}{1,2,3,6,8}A \cap C = \{1,2,3,\} \cap \{1,2,3,6,8\}

STEP 4

Calculate the intersection of sets A and C.
AC={1,2,3}A \cap C = \{1,2,3\}

STEP 5

Now that we have the intersection of sets A and C, we need to find its complement. The complement of a set is the set of all elements in the universal set that are not in the given set.
(AC)={xxUandx(AC)}(A \cap C)^{\prime} = \{x x \in U \, and \, x \notin (A \cap C)\}

STEP 6

Plug in the values for the universal set U and the intersection of sets A and C to find the complement.
(AC)={1,2,3,4,5,6,,8}{1,2,3}(A \cap C)^{\prime} = \{1,2,3,4,5,6,,8\} - \{1,2,3\}

STEP 7

Calculate the complement of the intersection of sets A and C.
(AC)={4,5,6,7,}(A \cap C)^{\prime} = \{4,5,6,7,\}The set (AC)(A \cap C)^{\prime} is {4,5,6,7,}.

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