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Various Problems in Discrete Mathematics

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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 Summary

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

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 ...

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