Concept explainers
In each of 28-31: a. Rewrite the theorem in three different ways: as
and as If_________, then________(without using an explicit
universal quantifier).
b. Fill in the blanks in the proof of the theorem.
Theorm: The sum of any two odd integers is even.
Proof: Suppose m and n are any [particulasr but arbitarity chosen] odd integers.
[We must show that m+n is even.]
By (a) m+2r+1 and n=2s+1 for some integers r and s.
Theorem 4,1-2: The sum of any even integer and jjiv odd integer u odd.
Proof: Suppose m la any even integer and n is
__(a)__By definition of even. m-2r for some
(b) and by definition of odd, n = 2s + 1 for
some integer s. By subsitution and slgebra.
Since r and s are both integers, so is their nun r + s. Hence m + n has the form twice some integer plus one and so (d)__by definition of odd.
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Check out a sample textbook solutionChapter 4 Solutions
Discrete Mathematics With Applications
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,