Question

which of the following describes the set {x | x > x + 1 and x ∈ w}?
a. it is not well defined. there is no clear meaning of the description \x ∈ w\.
b. it is well defined. it is the set of all natural numbers, n.
c. it is not well defined. there is no clear meaning of the description \x > x + 1\.
d. it is well defined. it is the set of all whole numbers, w.
e. it is not well defined. given a whole number x, it is not possible to tell whether x > x + 1 is true or false.
f. it is well defined. it is the empty set ∅.

Explanation:

Step1: Analyze the inequality \( x > x + 1 \)

Subtract \( x \) from both sides of the inequality \( x > x + 1 \). We get \( x - x > x + 1 - x \), which simplifies to \( 0 > 1 \).

Step2: Evaluate the truth of \( 0 > 1 \)

The statement \( 0 > 1 \) is false. For any whole number \( x \) (since \( x\in W \)), the inequality \( x > x + 1 \) will never hold because it leads to a false statement \( 0 > 1 \). So, there are no elements \( x \) in the set of whole numbers \( W \) that satisfy \( x > x + 1 \).

Step3: Determine the nature of the set

A set defined by a condition that no elements satisfy is the empty set. Also, the set is well - defined because we can clearly determine that no whole number satisfies \( x > x + 1 \).

Answer:

F. It is well defined. It is the empty set \( \varnothing \)