**Conditionals and Biconditionals
Logic is the hygiene the mathematician
practices to keep his ideas healthy and strong.
—**

** Definitions.** Given propositions

Many people have problems understanding the truth values for the conditional. The following example will help illustrate the truth values for the conditional.

A person makes the promise: "If I find a $20 bill, then I will take you to a movie." Now we consider when the person has broken the promise.

(1) The person finds
a $20 bill and takes you to a movie.

is true, since the person did not break the
promise. Note *p* and *q* are both true.

(2) The person finds
a $20 bill and does not take you to a movie.

is false, since the person broke the
promise. Note *p* is true, but *q* is false.

(3) The person did
not find a $20 bill, but took you to a movie anyway.

is true, since the person did not break the
promise. Note *p* is false,

but *q* is true. *(This is the case
with which many people have trouble understanding.)*

(4) The person did not find a $20 bill and did not take you to a movie.

is
true, since the person did not break the promise. Note *p* and *q
* are both false.

** Examples of true statements.**(a) If

(b) If

(c) If two lines are perpendicular to the same line, then the two lines are parallel.

(d) If 3 < 2, then 9 < 4.

(e) If 2 < 3, then 4 < 9.

(f) If 3 < 2, then 4 < 9.

(a) If

(b) If 2 < 3, then 9 < 4.

*Definitions.*

The *converse* of is .

The *inverse* of is .

The *contrapositive* of is .

*Examples.*

conditional: If a polygon is a square, then it has four sides of equal length.

converse: If a
polygon has four sides of equal length, then it is a square.

inverse: If a polygon is not a square,
then it does not have four sides of equal length.

contrapositive: If a polygon does not
have four sides of equal length, then it is not a square.

conditional: If *f * is differentiable at *a* and has a relative extremum at *a*, then *f '*(*a*) = 0.

converse: If *f '*(*a*) = 0, then *f* is differentiable at *a* and has a relative extremum at *a*.

inverse: If *f* is not differentiable at *a* or does not have a relative extremum
at *a*, then

contrapositive: If then *f * is not differentiable at *a* or does not have a relative extremum at *a*.

A conditional and its contrapositive are equivalent.

A conditional is not equivalent to either its inverse or its
converse. Note that the inverse of a conditional is the contrapositive of the
converse.

*Definition.**p* if and
only if *q* is a *biconditional* statement and is
denoted by and often written as *p* iff *q*. A biconditional is true only when *p* and *q* have the same
truth value. The truth table for the biconditional is

Note that is equivalent to

Biconditional statements occur frequently in mathematics. Definitions are usually biconditionals.

** Examples.**(a) A quadrilateral is a
rectangle if and only if it has four right angles.

** Theorem 1.** For propositions

(a) is logically equivalent to

(b)
is logically equivalent to .(*DeMorgan's*)

(c)
is logically equivalent to .
(*DeMorgan's*)

(d) is logically equivalent to .

(e) is logically equivalent to .

(f) is logically equivalent to .

(g) is logically equivalent to .

(h) is logically equivalent to .

** Examples.**(a) "It is false that the
transformation is an isometry and the transformation is a dilation." is
equivalent to "The transformation is not an isometry or the transformation
is not a dilation."

(b) "It is not the case that if John is 16 years old, then John has a driver's license." is equivalent to "John is 16 years old and does not have a driver's license."

** Definitions.
**Conditionals are often written in forms other than
if-then. The terms

"For *f* to be an
isometry it is sufficient that *f * is a rotation."

"If *f * is a rotation, then *f * is an isometry."

"For *f* to be a rotation, it is necessary that *f*
is an isometry."

"*f * is a rotation only if *f* is an isometry."

" For two triangles to be perspective from a point, it is necessary for the triangles to be perspective from a line."

"Two triangles are perspective from a point only if the
two triangles are perspective from a line*.*"

"If two triangles are perspective from a point, then they are perspective
from a line."

"For two triangles to be perspective from a line, it is sufficient that the
two triangles are perspective from a point."

© Copyright 2005, 2006 - Timothy Peil |