A logical statement is a sentence which is either true or false. "In ten years, the Prime Minister of the United Kingdom will be a woman" is a prediction. While its truth value will not be established for a while, it is either true or false, and it is therefore a logical statement.

A set is well-defined if it can be determined without ambiguity what elements are - and are not - in the set. Since every person either has lived at least two hundred years or has not lived at least two hundred years, this set meets the criterion for being well-defined.

Note that this set is (probably) the empty set, since no one alive is older than 200; the condition holds nonetheless.

This is an example of a conditional statement with a true antecedent and a true consequent, since  and 13 is a prime number. Even though the consequent does not result from the antecedent, the truth value of such a conditional is still "true".

The number of subsets of a set with elements is . It follows that the number of sets of any subset must be exactly a power of 2. 1,000 is not a power of 2, so there cannot be a set with exactly that many subsets.

The number of subsets of a set with elements is . It follows that the number of sets of any subset must be exactly a power of 2. Since

,

a set with 1,024 subsets can exist, and such a set would have ten elements. Of the given choices, the set

is such a set.

The statement

"Andy is a Freemason if and only if neither Benny nor Charlie is a Freemason."

is a biconditional statement. It can be rewritten by negating both parts of the statement as

"It is not true that Andy is a Freemason if and only if it is not true that neither Benny nor Charlie is a Freemason."

or, simplifying it a little,

"Andy is not a Freemason if and only if either Benny or Charlie is a Freemason."

Benny is a Freemason, making the second part of the latter biconditional statement true whether or not Charlie is a Freemason. It follows that Andy is not a Freemason. However, whether Charlie is a Freemason or not, his status is consistent with the biconditional.

We therefore know that Andy is not a Freemason, but we do not know whether or not Charlie is a Freemason.

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 , since, given any person in history, that person is/was or is/was not the President of the United States, is a well-defined set.

The selection of a license plate number can be seen as a series of independent events, as follows:

First, a prefix of one or two letters must be chosen. One way to look at this is that the prefix can be any one of 25 letters ("X" is excluded) followed by either any of 26 letters or a blank. By the multiplication principle, there are

possible prefixes.

The remaining characters must comprise five numeral; since there are no restrictions on the digits, by the multiplication principle, the number of possible numeral strings is

Applying the multiplication principle one more time, there will be

The selection of a license plate number can be seen as a series of independent events, as follows:

First, the positions of the two letters is chosen. Since this is a choice of two positions out of six, without regard to order - the number of combinations of two from a set of six - the number of ways to choose these positions is .

Next, the two letters are chosen. Repetition is allowed, and there are 24 letters from which to choose, so, by the multiplication principle, there are  ways to choose the letters.

Next, the four numerals are chosen. Repetition is allowed, and there are 10 digits from which to choose, so, by the multiplication principle, there are ways to choose the digits.

Applying the multiplication principle one more time, this gives us

different license plate numbers.

The negation of a statement can be stated as "It is not true that ." Therefore, the negation of the statement "" is the statement "It is not true that ." This can be restated as - not . Therefore, is not the negation of .