Question

let u = {all integers}.
consider the following sets:
a = {x | x ∈ u and x > 3}
b = {x | x ∈ u and x is an even integer}
c = {x | x ∈ u and 2x is an odd integer}
d = {x | x ∈ u and x is an odd integer}
use the defined sets to answer the questions.
assuming 0 is an even integer, which set is the complement to set b?
d
which set is an empty set?
c
which set would contain the subset e = {1, 3, 5, 7}?
a, b, c, d

Explanation:

Step1: Analyze the third question

We need to find which set contains the subset \( E = \{1, 3, 5, 7\} \). Let's analyze each set:

• Set \( A=\{x\mid x\in U \text{ and } x > 3\} \): The elements of \( E \) are \( 1, 3, 5, 7 \). Among these, \( 1\) and \( 3 \) are not greater than \( 3 \), so \( E \) is not a subset of \( A \).
• Set \( B=\{x\mid x\in U \text{ and } x \text{ is an even integer}\} \): The elements of \( E \) are odd integers, so \( E \) is not a subset of \( B \).
• Set \( C=\{x\mid x\in U \text{ and } 2x \text{ is an odd integer}\} \): For any integer \( x \), \( 2x \) is always even (since multiplying an integer by \( 2 \) gives an even number). So \( 2x \) can never be odd, so \( C \) is an empty set and cannot contain \( E \).
• Set \( D=\{x\mid x\in U \text{ and } x \text{ is an odd integer}\} \): The elements of \( E \) are all odd integers, so \( E \) is a subset of \( D \).

Step2: Correct the third question's answer

The original selected answer for the third question was \( A \), but based on the analysis, the correct set that contains \( E=\{1, 3, 5, 7\} \) is \( D \).

For the first question:

• The universal set \( U \) is all integers. Set \( B \) is the set of all even integers. The complement of a set of even integers in the set of all integers should be the set of all odd integers. Set \( D \) is the set of all odd integers, so the answer for the first question (complement of \( B \)) is correct as \( D \).

For the second question:

• Set \( C=\{x\mid x\in U \text{ and } 2x \text{ is an odd integer}\} \). As we know, for any integer \( x \), \( 2x \) is even (because \( 2x = 2\times x \), and by definition of even numbers, a number of the form \( 2k \) where \( k \) is an integer is even). So there are no integers \( x \) such that \( 2x \) is odd. So \( C \) is an empty set, and the answer for the second question is correct as \( C \).

Answer:

1. Complement of \( B \): \( D \) (correct as originally selected)
2. Empty set: \( C \) (correct as originally selected)
3. Set containing \( E = \{1, 3, 5, 7\} \): \( D \) (corrected from \( A \))