Studdy AI Logo
Math

QuestionFind (A∪B)′∩B(A \cup B)^{\prime} \cap B where U={1,2,…,17}U=\{1,2,\ldots,17\}, A={9,10,12,13,17}A=\{9,10,12,13,17\}, B={10,11,12,14}B=\{10,11,12,14\}.

Studdy Solution

STEP 1

Assumptions1. The universal set U is {1,,3,...,17} . Set A is {9,10,12,13,17}
3. Set B is {10,11,12,14}
4. We need to find the set (A∪B)′∩B(A \cup B)^{\prime} \cap B

STEP 2

First, we need to find the union of sets A and B, denoted as A∪BA \cup B. The union of two sets is a set containing all elements that are in A, or in B, or in both.
A∪B={xx∈A or x∈B}A \cup B = \{x x \in A \, or \, x \in B\}

STEP 3

Now, plug in the given values for sets A and B to calculate the union.
A∪B={9,10,12,13,17}∪{10,11,12,14}A \cup B = \{9,10,12,13,17\} \cup \{10,11,12,14\}

STEP 4

Calculate the union of sets A and B.
A∪B={9,10,11,12,13,14,17}A \cup B = \{9,10,11,12,13,14,17\}

STEP 5

Now that we have the union of sets A and B, we can find its complement, denoted as (A∪B)′(A \cup B)^{\prime}. The complement of a set is the set of all elements in the universal set that are not in the given set.
(A∪B)′={xx∈U and x∉(A∪B)}(A \cup B)^{\prime} = \{x x \in U \, and \, x \notin (A \cup B)\}

STEP 6

Plug in the values for the universal set U and the union of sets A and B to calculate the complement.
(A∪B)′={xx∈{1,2,3,...,17} and x∉{9,10,11,12,13,14,17}}(A \cup B)^{\prime} = \{x x \in \{1,2,3,...,17\} \, and \, x \notin \{9,10,11,12,13,14,17\}\}

STEP 7

Calculate the complement of the union of sets A and B.
(A∪B)′={1,2,3,4,5,6,7,,15,16}(A \cup B)^{\prime} = \{1,2,3,4,5,6,7,,15,16\}

STEP 8

Now that we have the complement of the union of sets A and B, we can find its intersection with set B, denoted as (A∪B)′∩B(A \cup B)^{\prime} \cap B. The intersection of two sets is a set containing all elements that are in both sets.
(A∪B)′∩B={xx∈(A∪B)′ and x∈B}(A \cup B)^{\prime} \cap B = \{x x \in (A \cup B)^{\prime} \, and \, x \in B\}

STEP 9

Plug in the values for the complement of the union of sets A and B and set B to calculate the intersection.
(A∪B)′∩B={xx∈{,2,3,4,5,6,7,8,15,16} and x∈{,11,12,14}}(A \cup B)^{\prime} \cap B = \{x x \in \{,2,3,4,5,6,7,8,15,16\} \, and \, x \in \{,11,12,14\}\}

STEP 10

Calculate the intersection of the complement of the union of sets A and B and set B.
(A∪B)′∩B=∅(A \cup B)^{\prime} \cap B = \emptysetThe intersection of the complement of the union of sets A and B and set B is the empty set.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord