Question

Which of the following statement is incorrect?

• A. All the null sets are equivalent to each other
• B. Two equal sets will always be equivalent sets
• C. Set of all real numbers and the set of integers are not equivalent sets
• D. Two equivalent sets will always be equal sets

Answer 1

The correct option is D Two equivalent sets will always be equal sets

All the null sets are equivalent to each other as they have the same cardinality.

If $A$ and $B$ are two sets such that $A=B,$ then $A$ is equivalent to $B$ as they have the same cardinality.
This means that two equal sets will always be equivalent but the converse of the same may or may not be true.
Eg: $A=\{1,-7,20.1,55\}$ and $B=\{a,2,3,d\},$ then $A$ is equivalent to $B$ but $A\ne B.$

Set of all real numbers and the set of integers are infinite sets, hence they are not equivalent sets.