1.     Let R(m, n) be the predicate “If m is a factor of n2 then m is a factor of n,” with domain for both m and n being Z the set of integers.
a. Explain why R(m, n) is false if m 5 25 and n 5 10.
b. Give values different from those in part (a) for which R(m, n) is false.
c. Explain why R(m, n) is true if m 5 5 and n 5 10.
d. Give values different from those in part (c) for which R(m, n) is true.

2.

3. 

4.

Write an informal negation for each of the follow-
2. Which of the following is a negation for “All dogs are loyal”? More than one answer may be correct.
a. All dogs are disloyal.
b. No dogs are loyal.
c. Some dogs are disloyal.
d. Some dogs are loyal.
e. There is a disloyal animal that is not a dog.

f. No athletic people are discrete mathematics students.

Be careful to avoid negations that are ambiguous.
a. All dogs are friendly.
b. All graphs are connected.
c. Some suspicions were substantiated.
d. Some estimates are accurate.

Rewrite the following statement informally in at least two different ways without using variables or the symbol Vor the words "for all." Vreal numbers x, if x is positive then the square root of x is positive. Rewrite the following statement informally in at least two different ways without using variables or the symbol Vor the words "for all." Vreal numbers x, if x is positive then the square root of x is positive.