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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.

0 3 Views | Posted 3 months ago
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    Answered by

    alok kumar singh | Contributor-Level 10

    3 months ago

    This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

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

    Let q: x be an integer and x2 be even

    Let r: x is even

    We have to prove, using the method of contra positive whether q⇒r is true.

    [i.e., its contra positive ∼r⇒∼q is true]

    Let ∼r be true i.e., r be false

    Let us assume that x is not even (integer) i.e., x is odd.

    ∴x=2m+1 where m is an integer

    ∴x2 (2m+1)2=4m+1+4m

    =4m2+4m+1

    =2 (2m2+2m+1) =2t+1    t=2m2+2m is an integer.

    ⇒ x2 is also odd ⇒ x is not even

    ⇒q   isfalse  [ Bydefof    q]⇒∼q istrue 

    ∴∼r⇒∼q is true

    Thus, by the method of contrapositive q⇒r is true

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