Which of the Following Statements Are True?
Which of the following statements are true? Give reason to support your answer.
(i) For any two sets \(A\) and \(B\) either \(A \subseteq B\) or \(B \subseteq A\).
(ii) Every subset of an infinite set is infinite.
(iii) Every subset of a finite set is finite.
(iv) Every set has a proper subset.
(v) \(\{a,b,a,b,a,b,\ldots\}\) is an infinite set.
(vi) \(\{a,b,c\}\) and \(\{1,2,3\}\) are equivalent sets.
(vii) A set can have infinitely many subsets.
Solution
(i) False
Example: \[ A=\{1,2\}, \quad B=\{2,3\} \] Here, \[ A \nsubseteq B \] and \[ B \nsubseteq A \]
(ii) False
An infinite set may have finite subsets. Example: \[ \mathbb{N} \] is infinite but \[ \{1,2,3\} \] is a finite subset.
(iii) True
Every subset of a finite set must also contain a finite number of elements.
(iv) False
The empty set \[ \emptyset \] has no proper subset.
(v) False
Repetition of elements does not change a set. Therefore, \[ \{a,b,a,b,a,b,\ldots\}=\{a,b\} \] which is a finite set.
(vi) True
Both sets have three elements. Hence they are equivalent sets.
(vii) True
Infinite sets can have infinitely many subsets. Example: \[ \mathbb{N} \] has infinitely many subsets.