TMUA Master

Consider the statement

If $x>3$, then $x^2>9$.

This is an implication. In general, an implication has the form

If P, then Q.

P is the assumption and Q is the conclusion. If the implication is true, then whenever P is true, Q must also be true. But knowing Q is true does not automatically tell you that P is true.

For example, “All multiples of $6$ are even” can be rewritten as “If an integer is a multiple of $6$, then it is even.” Writing a statement in If P, then Q form is often the easiest way to see its direction.

Reading If, Only If and If and Only If

The words if and only if give different directions.

For example,

$x^2<9$ if $-3<x<3$

means if $-3<x<3$, then $x^2<9$, whereas

$x^2<9$ only if $x<3$

means if $x^2<9$, then $x<3$.

An if and only if statement gives both directions. For example, $|x|<3$ if and only if $-3<x<3$.

If the direction is unclear, rewrite the statement in If P, then Q form before using it.

Converse and Contrapositive

For an implication If P, then Q, there are two related statements to distinguish.

For example, “If $x>4$, then $x^2>16$” is true. Its converse, “If $x^2>16$, then $x>4$”, is false because $x=-5$ satisfies $x^2>16$ but not $x>4$.

The contrapositive of the original statement is

If $x^2\leq16$, then $x\leq4$.

The contrapositive is useful when the question tells you that Q is false. For example, suppose a geometric sequence has first term $a>0$ and common ratio $r>0$. Since $\frac{u_6}{u_3}=r^3$,

When forming a contrapositive, negate the whole condition.

For example, “If $x>0$ and $y>0$, then $xy>0$” has contrapositive “If $xy\leq0$, then $x\leq0$ or $y\leq0$”.

More complicated negations are covered in Chapter 4.

Restrictions from the original statement still apply when you rewrite it, check its converse or use its contrapositive. If the statement concerns integers, positive values or values in a particular interval, use only values in that domain.

Summary

When using an implication:

1. Rewrite it as If P, then Q if the direction is unclear.
2. Do not reverse it unless the converse has been checked.
3. If the question tells you that Q is false, try the contrapositive If not Q, then not P.
4. Keep all original restrictions when you rewrite or test the statement.