Question

analyzing sets
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?
which set is an empty set?
which set would contain the subset e = {1, 3, 5, 7}?
options for the last question: a, b, c, d (with a selected)

Explanation:

Step 1: Complement of Set B

The universal set \( U \) is all integers. Set \( B \) is the set of even integers. The complement of \( B \), denoted \( B' \), is the set of elements in \( U \) not in \( B \). Since even and odd integers are disjoint and cover all integers, the complement of \( B \) (even integers) is the set of odd integers, which is set \( D \).

Step 2: Empty Set

For set \( C \): \( 2x \) is always even (since multiplying any integer by 2 gives an even number). An even number can never be odd. So there are no integers \( x \) such that \( 2x \) is odd. Thus, set \( C \) is empty.

Step 3: Subset \( E = \{1, 3, 5, 7\} \)

• Set \( A \): \( x > 3 \), but \( 1, 3 \) are not \( > 3 \), so \( E

ot\subseteq A \).

• Set \( B \): Even integers, but \( E \) has odd integers, so \( E

ot\subseteq B \).

• Set \( C \): Empty, so no subset.
• Set \( D \): Odd integers. \( 1, 3, 5, 7 \) are all odd, so \( E \subseteq D \).

Answer:

s:

1. Complement of \( B \): \( D \)
2. Empty set: \( C \)
3. Contains \( E \): \( D \)