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Zorluk: OrtaSets and Set Operations

Given the universal set U={x∈Z:1≤x≤15}\mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 15\}, with subsets P={x∈U:x is a prime number}P = \{x \in \mathcal{U} : x \text{ is a prime number}\} and Q={x∈U:x is an odd number}Q = \{x \in \mathcal{U} : x \text{ is an odd number}\}, what is the number of elements in the set (P∪Q)′(P \cup Q)'?

Cevap: 6 / six / 6 elements

Cevap

The number of elements in (P∪Q)′(P \cup Q)' is 6.
The set P∪QP \cup Q consists of all numbers from 1 to 15 that are either prime or odd: {1,2,3,5,7,9,11,13,15}\{1, 2, 3, 5, 7, 9, 11, 13, 15\}. The complement (P∪Q)′(P \cup Q)' relative to U\mathcal{U} contains all elements of U\mathcal{U} that are neither prime nor odd, which are the even composite numbers: {4,6,8,10,12,14}\{4, 6, 8, 10, 12, 14\}. Counting these elements gives 6.

Adım Adım Çözüm

1
List all elements of the universal set U\mathcal{U}, subset PP, and subset QQ.
U={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}\mathcal{U} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}, P={2,3,5,7,11,13}P = \{2, 3, 5, 7, 11, 13\}, and Q={1,3,5,7,9,11,13,15}Q = \{1, 3, 5, 7, 9, 11, 13, 15\}.
Listing the explicit elements allows precise execution of set union and complement operations.
2
Find the union P∪QP \cup Q.
P∪Q={1,2,3,5,7,9,11,13,15}P \cup Q = \{1, 2, 3, 5, 7, 9, 11, 13, 15\}.
The union combines all unique elements that belong to either set PP, set QQ, or both.
3
Determine the complement set (P∪Q)′(P \cup Q)' relative to U\mathcal{U} and count its elements.
(P∪Q)′={4,6,8,10,12,14}(P \cup Q)' = \{4, 6, 8, 10, 12, 14\}, which contains 6 elements.
The complement set consists of all elements in U\mathcal{U} that are not present in P∪QP \cup Q.

Anahtar Kavram

Complement of Set Union
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Konular

Sets and Set Operations

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