Concept explainers
Find the mistakes in the “proofs” show in 15-19.
Theorem: The product of any even integer and n is any odd integer is even.
“Proof: Suppose m is any even integer and n is any odd integer. If
Where r is an integer. By definition of even, then, m.n is even, as was to be shown.”
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Discrete Mathematics With Applications
- Prove that if p and q are distinct primes, then there exist integers m and n such that pm+qn=1.arrow_forwardProve that if and are real numbers such that , then there exist a rational number such that . (Hint: Use Exercise 25 to obtain such that . Then choose to be the least integer such that . With these choices of and , show that and then that .) If and are positive real numbers, prove that there exist a positive integer such that . This property is called Archimedean Property of the real numbers. (Hint: If for all , then is an upper bound for the set . Use the completeness property of to arrive at a contradiction.)arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,