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Suppose you have two numbers x, y ∈ Q (that is, x and y are rational...

Suppose you have two numbers x, y ∈ Q (that is, x and y are rational numbers).

a) Use the formal definition of a rational number to express each of x and y as the ratio of two integers. Remember, x and y could be different numbers!

(b) Use your results from (a) to show that xy must also be a rational number. Carefully justify your answer, showing how it satisfies the formal definition of a rational number.

(2) Suppose that a and b are integers.

(a) Use the formal definition of ”even integer” to translate the following statements: a is an even integer. b is an even integer. :. ab is an even integer.

(b) Using your answers from (a), write the statement ”If ab is an even integer, then a is an even integer or b is an even integer” using formal notation.

(c) ) What is the negation of your statement in (b)? Simplify as much as possible. (Keep in mind that if an integer is not even, then it must be odd! Also keep in mind the definition of an odd integer.)

(d) Show that the statement you wrote in (c) leads to a contradiction. Carefully show your algebra. What does this tell us about the original statement?

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