Answer to Question #350542 in Discrete Mathematics for Kenetha

Question #350542

Using proof by contraposition, show that if n is an integer and 5 added to its cube is odd then n is even. Before showing your solution, rewrite the statement to the proper form of a conditional statement then assign variables to the simple propositions. Show also the contrapositive form of the simple propositions before proceeding to your solution.

1
Expert's answer
2022-06-21T12:26:11-0400

Assign variables to the simple propositions:

"A(n) = "n^3 + 5 \\text{ is odd}"; \\\\\nB(n) = "n \\text{ is even}"."

Then we need to show that

"A(n) \\implies B(n)."

By contraposition it is equivalent to

"\\lnot B(n) \\implies \\lnot A(n)"

(contrapositive form), that is "if n is odd then n3 + 5 is even".

Let n be odd. Then n3 is also odd (because if n doesn't have 2 as a divisor, then n3 doesn't as well). Hence n3 + 5 is even as the sum of two odd numbers. Q. E. D.


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