# Various Problems in Discrete Mathematics

Prove Each Directly.

1. The product of any two even integers is even.

Prove by cases, where n is an arbitrary integer and Ixl denotes the absolute

value of x.

2. [-x]=[x] (*Brackets are the x's is the absolute value symbol)

Give a counterexample to disprove each statement, where P(x) denotes an

arbitrary predicate.

3. Every month has exactly 30 days.

Let a, b, and c be any real numbers. Then a < b if and only if there

is a positive real number x such that a + x - b. Use this fact to prove

each.

4. If a+c is less than b+c, then a is less than b

Determine if the given sets are equal.

5. {x, {y}}, {{x},y}

Let A = {a, e, f, g,i}, B = {b, d, e, g, h}, C = {d, e, f, h, i}, and U= {a,b,... ,k}.

Find each set.

6. (A U B)'

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#### Solution Preview

1. Every even integer is divisible by two. Thus the product of two even integers is divisible by 4 and hence is divisible by 2 and is thus even.

2. If x is negative, then -x is ...

#### Solution Summary

We solve six simple problems in discrete mathematics. The product of any two even integers being even is determined.