QuestionQuestion

1.    Give a counterexample to disprove each of the following:
a. For all real numbers a and b, if a2 = b2 then a = b
b. For all integers m and n, if 2m + n is odd then m and n are odd

2. Prove that the difference of two odd integers is even. Give a justification at each step.

3. Prove that the sum of any two rational numbers is a rational number. Give a justification at each step.

4. Prove by contradiction that there is no greatest integer.

5. Prove by contraposition that for all integers n, if n2 is even then n is even.

6. Write the following sentences using predicate symbols and quantifiers, and then negate the statements.
a. All programmers enjoy discrete mathematics
b. Some integers are not odd
c. Every integer that is divisible by 2 is even
d. There exists a natural number that is not a positive integer

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