DISCRETE MATH

1(a). Assume that t is a particular integer. Use the definition of even, odd, or composite to justify the following.

(a) 35 is an odd integer.

(b) 14 is an even integer.

(c) 77 is a composite integer.

(d) 14t 8 is an even integer.

(e) 8t + 9 is an odd integer.

1(b). For all real numbers x, x^2 > 0.

(a) Write the negation of the statement above.

(b) Disprove the original statement by giving a counterexample.

1(c). Prove: There is an integer n such that 7n^2 5n + 11 is a prime number.

1(d). The sum of any two green numbers is a blue number.

(a) Rewrite the statement with variables and in an "if...then..." form.

(b) What are the hypotheses of the statement?

(c) What is the conclusion of the statement?

(d) Prove the statement.

1(e). For all integers n, if n is blue, then n^2 is green.

(a) What are the hypotheses of the statement?

(b) What is the conclusion of the statement?

(c) Prove the statement.

1(f). Write each number as a ratio of two integers.

(a) 6.2893

(b) 2/7 + 4/3

(c) 0.243243243243243243 . . .

(d) 562.93045045045045045045045 . . .

1(g). Assume that k is an integer. Explain why 5k / k^2 + 2 must be a rational number.