Book of Proof - Amazon S3

36 Logic
2.2 And, Or, Not
The word “and” can be used to combine two statements to form a new
statement. Consider for example the following sentence.
R 1 : The number 2 is even and the number 3 is odd.
We recognize this as a true statement, based on our common-sense understanding
of the meaning of the word “and.” Notice that R 1 is made up of
two simpler statements:
P : The number 2 is even.
Q : The number 3 is odd.
These are joined together by the word “and” to form the more complex
statement R 1 . The statement R 1 asserts that P and Q are both true. Since
both P and Q are in fact true, the statement R 1 is also true.
Had one or both of P and Q been false, then R 1 would be false. For
instance, each of the following statements is false.
R 2 : The number 1 is even and the number 3 is odd.
R 3 : The number 2 is even and the number 4 is odd.
R 4 : The number 3 is even and the number 2 is odd.
From these examples we see that any two statements P and Q can
be combined to form a new statement “P and Q.” In the spirit of using
letters to denote statements, we now introduce the special symbol ∧ to
stand for the word “and.” Thus if P and Q are statements, P ∧Q stands
for the statement “P and Q.” The statement P ∧Q is true if both P and Q
are true; otherwise it is false. This is summarized in the following table,
called a truth table.
P Q P ∧Q
T T T
T F F
F T F
F F F
In this table, T stands for “True,” and F stands for “False.” (T and F are
called truth values.) Each line lists one of the four possible combinations
or truth values for P and Q, and the column headed by P ∧Q tells whether
the statement P ∧Q is true or false in each case.