Question

Show that the following statement is true by the method of the contrapositive 

p : “If x is an integer and x² is odd, then x is also odd.”

Answer 1

Let us Assume that q and r be the statements given 

q: x is an integer and x² is odd. 

r: x is an odd integer. 

since the given statement can be written as : 

p: if q, then r. 

Let r be false . then, 

x is not an odd integer, then 

x is an even integer 

x = (2n) for some integer n 

x² = 4n² 

x² is an even integer 

Thus, q is False 

Therefore, r is false ⇒ q is false 

Hence, p: “ if q, then r” is a true statement.