If-then statements
April 15, 2008
What is an if-then statement?
• One of the postulates we looked at earlier
stated:
If B is between A and C, then AB + BC =
AC.
• This type of statement is known as an ifthen statement for obvious reasons. We
also refer to them as conditional
statements or just conditionals.
All about conditionals
• A conditional consists of two parts; a
hypothesis and a conclusion.
• The hypothesis is the part of the statement
immediately after the “if,” which we will
represent by the letter p.
• The conclusion is the part of the statement
after the “then,” which we will represent by
the letter q.
Writing if then statements using
p and q
• You can write the conditional statement, “if
p then q” like this:
p→q
What is the converse?
• The converse of a conditional is formed by
interchanging the hypothesis and the
conclusion.
• So, instead of if p then q, the converse is if
q then p.
• We can write this as
q→p
If the conditional is true, is the
converse true?
• If you are in Ameena’s math class, then
you attend CMC.
• What is the converse?
• To prove a conditional statement false all
we need to do is find one counterexample.
What if both the conditional and
the converse are true?
• If both a conditional and its converse are
true (we never said it wasn't possible!) we
can combine them into a single statement
using the words "if and only if."
What is the inverse?
• If we negate the hypothesis and the
conclusion we get the inverse.
• Instead of if p, then q, we have if not p,
then not q.
• We can write this as:
~p → ~q
What is the contrapositive?
• Finally, if we interchange and negate the
hypothesis and conclusion we get the
contrapositive.
• The contrapositive of "If p, then q" is "If not
q, then not p.“
• We write this as:
~q→~p
What is logical equivalence?
• When two statements always have the
same truth value we say that they are
logically equivalent.
• So a conditional and its contrapositive are
logically equivalent as are the converse
and the inverse.
• .
What kinds of logical
equivalence are possible?
• All four may be true or all four may be
false.
• Two of the statements might be true and
two might be false but these are the only
possibilites.
• It is not possible for three of the
statements to be true and the last one
false or for three of the statements to be
false and the last one true.
To review
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Conditional: p→q
converse: q→p
inverse: ~p→~q
contrapositive: ~q→~p