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Exercises 1.4.7 Exercises
1.
Give the definition for these terms. Be sure to set up any notation that is required.
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2.
Use a direct proof to show that the sum of two odd integers is even.
3.
Use a direct proof to show that the sum of two even integers is even.
4.
Use a direct proof to show that the product of two odd integers is odd.
5.
Use a proof by contradiction to prove that the sum of an irrational number and a rational number is irrational.
6.
Use a direct proof to show the product of two rational numbers is a rational number.
7.
Prove each of these statements is false by providing a counterexample.
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The product of two irrational numbers is irrational.
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The sum of two irrational numbers is irrational.
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Every positive integer can be written as the sum of the squares of three integers.
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If
\(n\) is an integer, then
\(n^2 \gt n\text{.}\)
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If
\(a\) and
\(b\) are rational numbers, then
\(a^b\) is also rational.
8.
Let
\(P(n)\) be the proposition “
\(n^2 \geq n\text{.}\)” Prove
\(P(1)\) is true.
9.
Use a proof by contradiction to prove that if
\(n\) is an integer and
\(n^3 + 5\) is odd, then
\(n\) is even.
10.
Prove there is a right triangle with all three sides having rational lengths.
11.
Prove there exists a pair of consecutive integers such that one of these integers is a perfect square and the other is a perfect cube.
12.
Use a proof by contraposition to prove that if
\(n\) is an integer and
\(n^3 + 7\) is even, then
\(n\) is odd.
13.
