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Validity of Statements

To find the truth value of compound statements connected with “and”, “or” and “if-then”, we follow the following rules:-
  1. If p and q are statements, p and q is true if
    1. p is true
    2. q is true
So, show that the statement p is true and also show that q is true.  
  1. If p and q are statements, such that p or q is given, then
    1. Assuming p is false, show that q is true
    2. Assuming q is false, show that p is true
  2. Statement with if then
    1. Assuming p is true, prove q must be true (direct method)
    2. By assuming q is false, prove p must be false (contra positive method).
  3. Statements with “if and only if”. We need to show.
    1. if p is true, then q is true.
    2. if q is true, then p is true.
Example 1:
Check whether the given statement is true or false. If such that are even, then is even.

, x is even and y is even
is odd.
Let us apply Rule 3 (a),
p is true means x and y are even numbers
Then which is even
q is true
The statement is true.

Example 2:
By contradiction, check the truth value of the statement, is irrational

Assume, is rational
Then (are rationals, with no common factors)
Which contradicts our assumption that there is no common factor for a and b.
is irrational.

Example 3:
By giving a counter example, show that the following statement is false.
“If n is a prime number, it has to be odd.”

p: n is a prime number
q: n is odd.
Let us look for an even number which is prime. Obviously only '2' is an even prime.
p: n is a prime number
~ q: n is even
p ~q is true
p q is false.

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