of q is “it is false that q ”, Thus ∼ q is the statement. ∼ q: There does not exist a rational number x such that x = . This statement can be rewritten as ∼ q: For all real numbers x , x ≠ (iii) The negation of the statement is ∼ r: There exists a bird which have no wings. MATHEMATICS (iv) The negation of the given statement is ∼ s: There exists a student who does not study mathematics at the elementary level.
Example Using the words “necessary and sufficient” rewrite the statement “The integer n is odd if and only if n is odd”. Also check whether the statement is true. Solution The necessary and sufficient condition that the integer n be odd is n must be odd. Let p and q denote the statements p : the integer n is odd.
q : n is odd. To check the validity of “ p if and only if q”, we have to check whether “if p then q ” and “if q then p ” is true. Case If p , then q If p , then q is the statement: If the integer n is odd, then n is odd. We have to check whether this statement is true.
Let us assume that n is odd. Then n = k + when k is an integer. Thus n = ( k + ) = k + k + Therefore, n is one more than an even number and hence is odd. Case If q , then p If q , then p is the statement If n is an integer and n is odd, then n is odd.
We have to check whether this statement is true. We check this by contrapositive method. The contrapositive of the given statement is: If n is an even integer, then n is an even integer n is even implies that n = k for