Question

which of the following sets are not well defined? explain.
a. the set of short programmers
b. the set of moving poems
c. the set of whole numbers greater than or equal to 155
d. the set of subsets of {a,b,c,d,e,f}
e. the set {x | x > x + 2 and x ∈ w}

which of the following describes the set {x | x > x + 2 and x ∈ w}?
a. it is not well defined. there is no clear meaning of the description \x > x + 2\
b. it is not well defined. given a whole number x, it is not possible to tell whether x > x + 2 is true or false
c. it is not well defined. there is no clear meaning of the description \x ∈ w\
d. it is well defined. it is the empty set. ∅
e. it is well defined. it is the set of all whole numbers, w
f. it is well defined. it is the set of all natural numbers, n

Explanation:

For the first question (Which sets are not well - defined?):

• Set a (short programmers): The term "short" is subjective. There is no clear - cut definition of how short a programmer needs to be to be in this set. So, we can't precisely determine the elements of this set.
• Set b (moving poems): The concept of a "moving poem" is ambiguous. Different people may have different interpretations of what a moving poem is, so we can't clearly identify the elements of this set.
• Set e ($\{x|x > x + 2,x\in W\}$): For any whole number $x$, $x+2$ is always greater than $x$ (since $x+2-x = 2>0$). So, there are no elements in this set that satisfy the condition, and it is an empty set. But more importantly, from the perspective of set - building, the condition $x>x + 2$ for whole numbers is a contradiction, and it is not a well - defined way to describe a set in the sense that the condition is impossible to satisfy for the given domain (whole numbers).

Sets a, b, and e are not well - defined.

We know that for any whole number $x$, if we consider the inequality $x>x + 2$, we can subtract $x$ from both sides of the inequality. We get $0>2$, which is a false statement. So, there are no whole numbers $x$ that satisfy the inequality $x>x + 2$. This means that the set $\{x|x > x + 2,x\in W\}$ has no elements, i.e., it is the empty set $\varnothing$.

Answer:

a. The set of short programmers, b. The set of moving poems, e. The set $\{x|x > x + 2,x\in W\}$

For the second question (Which describes the set $\{x|x > x + 2,x\in W\}$?):