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.

A set is well-defined if each element can be identified with certainty as being or not being an element of the set. Since, for any value of , it can be clearly determined through substitution whether or not it is a solution of the given equation, is indeed well-defined.

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.

Call the hypothesis of the conditional, "", and call the conclusion, "". Then the given statement is the conditional "If then ."

The inverse of this conditional is the conditional "If (not ) then (not )/"- the conditional which negates the antecedent and the consequent,

"Not " is the negation of "", which is "It is not true that ", or, restated, "". Similarly, "Not " is the statement .

Thus, the inverse of the conditional is

"If , then ."

The negation of a statement can be stated as (not ), or, "It is not true that ." Therefore, the negation of the statement "The horse is black" is "It is not true that the horse is black" - or, restated, "The horse is not black." Since not being black is not the same as being white - the horse could be brown, for example - then is not the negation of .

denotes the intersection of and , the set of all elements the two sets share. Inspect the two sets, whose shared elements are in red:

.

It follows that

.

denotes the union of and , the set of all elements that are in , , or both. To find this set, first, collect the elements in both sets:

 and :

The union, including duplicates, is

Arrange, and eliminate the duplicates, shown in red:

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 negation of a logical proposition is the proposition "Not ," or, "It is not true that ."

The negation of the proposition " can be found by first noting that the solution set of that arithmetic inequality can be found to be the set . Therefore, the proposition "" is equivalent to the given proposition. The negation of this statement is "It is not true that ," or, equivalently, the proposition ""

, the complement of , is the set of all elements of not in :

A set with elements has subsets; since the set itself, which is not considered a proper subset, is one of those sets, the number of proper subsets is . has 4 elements, so, setting , has

proper subsets.