Q22 of 21 Page 14

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

p: If x is an integer and x2 is even, then x is also even.

Let q and r be the statements given by:


q: If x is an integer and x2 is even


r: x is an even integer.


Then, p: “If q, then r”.


If possible, let r be false. Then,


⇒ x is not an even integer.


⇒ x is an odd integer.


⇒ x = (2m + 1) for some integer m.


⇒ x2 = (2m + 1)2


⇒ x2 = 4m2 + 1 + 4m


⇒ x2 = 4m (m + 1) + 1


⇒ x2 is an odd integer.


⇒ q is false.


Thus, r is false


⇒ q is false.


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


More from this chapter

All 21 →
20

Check validity of the following statements.
(i) p: 125 is divisible by 5 and 7.
(ii) q: 131 is a multiple of 3 or 11.


 21

Prove the following statement by contradiction method.
p: The sum of an irrational number and a rational number is irrational.

 23

Form the biconditional statement p⟷q, where
(i) p: The unit digits of an integer is zero.
q: It is divisible by 5.
(ii) p: A natural number is odd.
q: Natural number is not divisible by 2.
(iii) p: A triangle is an equilateral triangle.
q: All three sides of a triangle are equal.

 24

Rewrite each of the following statements in the form of conditional statements.
(i) The square of an odd number is odd.
(ii) You will get a sweet dish after the dinner.
(iii) You will fail, if you will not study.
(iv) The unit digit of an integer is 0 or 5, if it is divisible by 5.
(v) The square of a prime number is not prime.
(vi) 2b = a + c, if a, b and c are in AP.