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:

The union, including duplicates, is

Arrange, and eliminate the duplicates, shown in red:

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 8 elements, so, setting ,  has 

subsets.

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

The negation of the proposition "There is a man in the car" is "It is not true that there is a man in the car." This is synonymous with "There is not a man in the car."

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.

Another way to express and is as follows:

and

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

The only choice among the five that is not divisible by 3 is 64, so 64 is the correct choice.

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

The negation of "Wilson fought for the Confederacy during the Civil War." is "It is not true that Wilson fought for the Confederacy during the Civil War," or, more simply, "Wilson did not fight for the Confederacy during the Civil War." This is not synonymous with "Wilson fought for the Union during the Civil War"; for example, if Wilson wasn't born yet during the Civil War, the first proposition is true, but the second is false.

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.

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.

For any whole numbers , where ,

Setting :

.

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.