Question

which of the following sets are not well defined? explain.
a. the set of rich engineers
b. the set of moving books
c. the set of natural numbers greater than 95
d. the set of subsets of {a,b,c,d,e,f}
e. the set {x | x > x + 1 and x ∈ w}

c. which of the following describes the set of natural numbers greater than 95?
a. it is not well defined. it is not possible to list all of the natural numbers greater than 95
b. it is well defined. given a number, it is possible to tell whether it is a natural number greater than 95
c. it is not well defined. there is no clear meaning of \greater than 95\
d. it is not well defined. there is no clear meaning of
atural numbers\
e. it is well defined. given a number, it is possible to tell whether it is a natural number greater than 95
f. it is well defined. it is possible to list all of the natural numbers greater than 95

Explanation:

Part 1: Identifying Not Well - Defined Sets

A well - defined set has a clear rule to determine if an element belongs to it.

• Set a (The set of rich engineers): The term "rich" is subjective. There is no clear - cut definition of how much wealth makes an engineer "rich". So, we cannot precisely determine which engineers belong to this set.
• Set b (The set of moving books): The term "moving" is ambiguous. We don't have a clear definition of what it means for a book to be "moving" (e.g., moving at what speed, in what way). So, we can't clearly identify which books are in this set.
• Set e (The set \(\{x|x > x + 1\) and \(x\in\mathbb{W}\}\)): For any whole number \(x\), the inequality \(x>x + 1\) can be rewritten as \(0>1\) (by subtracting \(x\) from both sides), which is a false statement. So, there are no elements in this set, but the rule to determine membership is clear (we can check for any whole number \(x\) if \(x>x + 1\) is true). So, this set is well - defined (it's an empty set).
• Set c (The set of natural numbers greater than 95): Natural numbers are positive integers (or non - negative integers depending on the definition). For any number, we can clearly check if it is a natural number and if it is greater than 95. For example, 96 is a natural number and \(96>95\), so 96 belongs to the set; 95 is a natural number but \(95 = 95\) (not greater than 95), so 95 does not belong to the set. So, this set is well - defined.
• Set d (The set of subsets of \(\{a,b,c,d,e,f\}\)): The subsets of a given set can be clearly identified. The number of subsets of a set with \(n\) elements is \(2^{n}\), and we can list out all the possible subsets (e.g., \(\varnothing\), \(\{a\}\), \(\{b\}\), \(\{a,b\}\), etc.). So, this set is well - defined.

So, the sets that are not well - defined are the set of rich engineers (a) and the set of moving books (b).

Part 2: Analyzing the Set of Natural Numbers Greater than 95

• A set is well - defined if there is a clear rule to determine whether an element belongs to the set.
• For the set of natural numbers greater than 95, the rule is: an element \(x\) belongs to the set if and only if \(x\) is a natural number and \(x>95\).
• Given any number, we can check two things: first, if it is a natural number (for example, 96 is a natural number, \(- 1\) is not); second, if it is greater than 95 (for example, 96>95, 95 is not greater than 95). So, we can clearly determine the membership of any number in this set.
• The option that correctly describes this set is B: "It is well defined. Given a number, it is possible to tell whether it is a natural number greater than 95". Option A is wrong because we don't need to list all elements (and we can't list all natural numbers greater than 95 as there are infinitely many, but the set is still well - defined). Options C and D are wrong because the terms "natural numbers" and "greater than 95" have clear meanings. Option E is wrong because we can't list all natural numbers greater than 95 (there are infinitely many), but the set is well - defined regardless of being able to list all elements.

Answer:

s

1. The sets that are not well - defined are:

• a. The set of rich engineers
• b. The set of moving books

1. For the set of natural numbers greater than 95, the answer is B.