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And, Or, Not 37
Statements can also be combined using the word “or.” Consider the
following four statements.
S 1 : The number 2 is even or the number 3 is odd.
S 2 : The number 1 is even or the number 3 is odd.
S 3 : The number 2 is even or the number 4 is odd.
S 4 : The number 3 is even or the number 2 is odd.
In mathematics, the assertion “P or Q” is always understood to mean that
one or both of P and Q is true. Thus statements S 1 , S 2 , S 3 are all true,
while S 4 is false. The symbol ∨ is used to stand for the word “or.” So if P
and Q are statements, P ∨Q represents the statement “P or Q.” Here is
the truth table.
P Q P ∨Q
T T T
T F T
F T T
F F F
It is important to be aware that the meaning of “or” expressed in
the above table differs from the way it is sometimes used in everyday
conversation. For example, suppose a university official makes the following
threat:
You pay your tuition or you will be withdrawn from school.
You understand that this means that either you pay your tuition or you
will be withdrawn from school, but not both. In mathematics we never use
the word “or” in such a sense. For us “or” means exactly what is stated
in the table for ∨. Thus P ∨ Q being true means one or both of P and Q
is true. If we ever need to express the fact that exactly one of P and Q is
true, we use one of the following constructions:
P or Q, but not both.
Either P or Q.
If the university official were a mathematician, he might have qualified
his statement in one of the following ways.
Pay your tuition or you will be withdrawn from school, but not both.
Either you pay your tuition or you will be withdrawn from school.