denotes the Cartesian product of and , the set of all ordered pairs comprising an element of followed by an element of . The number of elements in is equal to the product of the numbers of elements in and :

.

The number of subsets of a set with elements is , so has subsets total. The statement is true.

Consider the second conditional "If Oscar is not a Freemason, then Nelson is not a Freemason.". It is known that Nelson is a Freemason, making the consequent of this conditional false. By a modus tollens argument, it follows that the antecedent is also false, and Oscar is a Freemason.

No conclusion can be drawn about Mickey, however. If Mickey is a Freemason, then by the first conditional, it follows that Nelson is a Freemason, which is already known; if Mickey is not a Freemason, no conclusion can be drawn that is inconsistent with what is known. Thus, either status is consistent with Nelson and Oscar both being Freemasons.

For any whole numbers , where ,

Setting :

.

A set is well-defined if it is clear which elements are in the set and which elements are not. There is a clear distinction between the people who died in 1941 and people who did not, so is well-defined.

The converse of a conditional statement reverses the antecedent and the consequent; the inverse negates both. That is, for a conditional

If , then ,"

the converse of the statement is

"If , then ,"

and the inverse is

"If (not ), then (not )."

This problem becomes easier if you know that the converse and the inverse of any conditional are logically equivalent - that is, one is true if and only the other is. It suffices to determine the truth value of one of them. The converse of the given conditional is the statement

"If , then ."

This is easily proved true; if , then , as proved through some algebra, and, by substitution, . Since the converse is true, the inverse is also true.

The converse of a conditional statement reverses the antecedent and the consequent; the contrapositive reverses and negates both. That is, for a conditional

If , then ,"

the converse of the statement is

"If , then "

and the contrapositive of the statement is

"If (not ), then (not )."

The converse of the given conditional is

"If , then ."

This is false, since this quadratic equation has two solutions - and - so if , it does not follow that .

The contrapositive of the conditional is

"If , then ."

This is a bit harder to prove, but it can be made easier by remembering that the contrapositive is actually logically equivalent to the original conditional - that is, one is true if and only the other is. The original statement

"If , then ,"

is true, as can easily be proved through substitution in the latter equation. It follows that the contrapositive is also true.

A set is well-defined if it can be determined with no ambiguity which elements are and are not in the set. In the case of , the word "smart" is ambiguous, since the definition can change according to who is deciding who is "smart" and who is not. is therefore not a well-defined set.

denotes the Cartesian product of and , the set of all ordered pairs comprising an element of followed by an element of . The number of elements in is equal to the product of the numbers of elements in and :

and , so .

Therefore, must be a set with two elements; of the choices, only fits that description.

A logical statement is a sentence that can be determined to be true or false. 7 is known to not be a composite number, so the sentence is known to be false; that makes it a valid example of a logical statement.

Given a conditional

"If , then ",

the converse, inverse, and contrapositive are, respectively:

Converse: "If , then ."

Inverse: "If (Not ), then (Not )",

Contrapositive: "If (Not ), then (Not )",

Let and be the antecedent and the conclusion of the given conditional - that is,

:

:

Then the statements and are the negations of and ; that is, they are (Not ) and (Not ), respectively. "If , then " is the conditional "If (Not ), then (Not )", making it the contrapositive of the original statement.