Discrete Mathematics Demystified

CHAPTER 1 Logic 13
is the statement
If x has four legs then x is a healthy horse
Notice that these statements have very different meanings: the first statement is true
while the second (its converse) is false. For instance, a chair has four legs but it is
not a healthy horse. Likewise for a pig.
EXAMPLE 1.18
The statement
If x > 5 then x > 3
is true. Any number that is greater than 5 is certainly greater than 3. But the converse
If x > 3 then x > 5
is certainly false. Take x = 4. Then the hypothesis is true but the conclusion is
false.
The statement
is a brief way of saying
A if and only if B
If A then B and If B then A
We abbreviate A if and only if B as A ⇔ B or as A iff B. Now we look at a truth
table for A ⇔ B.
A B A ⇒ B B ⇒ A A ⇔ B
T T T T T
T F F T F
F T T F F
F F T T T
Notice that we can say that A ⇔ B is true only when both A ⇒ B and B ⇒ A are
true. An examination of the truth table reveals that A ⇔ B is true precisely when
A and B are either both true or both false. Thus A ⇔ B means precisely that A and
B are logically equivalent. One is true when and only when the other is true.