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Let [tex]\( X = \{0, 1, 2, 3, 4, 5\} \)[/tex], [tex]\( Y = \{2, 4, 6, 7\} \)[/tex], and the Universe [tex]\( U = \{0, 1, 2, 3, 4, 5, 6, 7, 8\} \)[/tex].

Find [tex]\(\left(X^{\prime} \cup Y^{\prime}\right) \cap X\)[/tex].

A. [tex]\(\{0, 1, 3, 5\}\)[/tex]
B. [tex]\(\{0, 1, 2, 3, 4, 5\}\)[/tex]
C. [tex]\(\{2, 4\}\)[/tex]



Answer :

GinnyAnswer
To solve the problem, we need to find [tex]\(\left(X^{\prime} \cup Y^{\prime}\right) \cap X\)[/tex]. Let’s go step-by-step through the process:

1. Define Sets [tex]\(X\)[/tex], [tex]\(Y\)[/tex], and Universe [tex]\(U\)[/tex]:
[tex]\[ X = \{0, 1, 2, 3, 4, 5\} \][/tex]
[tex]\[ Y = \{2, 4, 6, 7\} \][/tex]
[tex]\[ U = \{0, 1, 2, 3, 4, 5, 6, 7, 8\} \][/tex]

2. Find the Complements [tex]\((X')\)[/tex] and [tex]\((Y')\)[/tex] with respect to the Universe [tex]\(U\)[/tex]:
- The complement of [tex]\(X\)[/tex], denoted [tex]\(X'\)[/tex], consists of all elements in the Universe [tex]\(U\)[/tex] that are not in [tex]\(X\)[/tex]:
[tex]\[ X' = U - X = \{0, 1, 2, 3, 4, 5, 6, 7, 8\} - \{0, 1, 2, 3, 4, 5\} = \{6, 7, 8\} \][/tex]
- The complement of [tex]\(Y\)[/tex], denoted [tex]\(Y'\)[/tex], consists of all elements in the Universe [tex]\(U\)[/tex] that are not in [tex]\(Y\)[/tex]:
[tex]\[ Y' = U - Y = \{0, 1, 2, 3, 4, 5, 6, 7, 8\} - \{2, 4, 6, 7\} = \{0, 1, 3, 5, 8\} \][/tex]

3. Calculate the Union of Complements [tex]\(X'\)[/tex] and [tex]\(Y'\)[/tex]:
[tex]\[ X' \cup Y' = \{6, 7, 8\} \cup \{0, 1, 3, 5, 8\} = \{0, 1, 3, 5, 6, 7, 8\} \][/tex]

4. Find the Intersection of [tex]\((X' \cup Y')\)[/tex] with [tex]\(X\)[/tex]:
[tex]\[ (X' \cup Y') \cap X = \{0, 1, 3, 5, 6, 7, 8\} \cap \{0, 1, 2, 3, 4, 5\} \][/tex]
The intersection consists of elements that are present in both sets:
[tex]\[ (X' \cup Y') \cap X = \{0, 1, 3, 5\} \][/tex]

So, the result is:
[tex]\[ \left(X' \cup Y'\right) \cap X = \{0, 1, 3, 5\} \][/tex]

Therefore, the answer is [tex]\(\{0, 1, 3, 5\}\)[/tex].

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