For any whole numbers , where ,

Setting :

.

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.

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.

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 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.

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.

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.

Let L stand for a letter and N stand for a numeral. One of two events will happen - the selection of a license plate with the pattern LNLNLNL, or the selection of a license plate with pattern NLNLNLN. These events are mutually exclusive, so we can count the number of ways to obtain them separately, then add.

There are no restrictions as to which letters or numerals can be chosen, or how many times each can be chosen, so the number of ways to choose a license plate number with the pattern LNLNLNL is

.

The number of ways to choose a license plate number with the pattern NLNLNLN is

.

Add these to get

,

the total number of license plates possible.

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