SAT SUBJECT TEST MATH LEVEL 1

MISCELLANEOUS TOPICS

CHAPTER 19
Logic

CONDITIONAL STATEMENTS

A conditional statement is a sentence of the form “if p, then q” where p and q are given statements. In logic, this is written p  q and is read “p implies q.”

For example:

If p is the statement “x is divisible by 4,” and q is the statement “x is even,” then p  q is the statement “If x is divisible by 4, then x is even.”

There are three conditional statements related to the statement p  q.

The converse of p  q is q  p
(the order of the statements is reversed).

The inverse of p  q is pq
(the original statements are negated).

The contrapositive of p  q is qp
(the order of the statements is reversed and the statements are negated).

For example:

Conditional: p  q:
If x is divisible by 4, then x is even.

Converse: q  p:
If x is even, then x is divisible by 4.

Inverse: pq:
If x is not divisible by 4, then x is not even.

Contrapositive: qp:
If x is not even, then x is not divisible by 4.

Note that in the preceding example, the conditional and its contrapositives are both true. Whether a conditional statement is true or false, its contrapositive always has the same truth value.

Key Fact R4

• A conditional statement is logically equivalent to its contrapositive. Either both statements are true or both are false.

• The converse and inverse of a conditional statement are logically equivalent. Either both statements are true or both are false.

The logic question on a Math 1 test often asks which of five statements is equivalent to a given conditional statement. The correct answer is always the contrapositive of the given conditional.

EXAMPLE 1: A statement equivalent to the conditional, “If Adam lives in California, then he lives in the United States” is its contrapositive, “If Adam does not live in the United States, then he does not live in California.”