Determine the truth value of each of the following implications.  a. “If 45 is divisible by 10, then 45 is also divisible by 5.”  b. “If 45 is divisible by 5, then 45 is also divisible by 10.”

Suppose you want to refute the statement: There are some natural numbers a, b, and c where a | b and b | c but a does not divide c.  Which of the following would be an appropriate strategy for you to employ? Group of answer choices Show specific values for a, b, and c where a | b and b | c and a | c. Show specific values for a, b, and c where a does not divide b and b does not divide c, but a | c. Show for any generic choices of a, b, and c that if a | b and b | c then a | c. Show for any generic choices of a, b, and c if a does not divide b and b does not divide c then a does not divide c.

Discrete mathematics. Write a formal proof for the following statement.

The following problem is that of Discrete Mathematics.  Consider the following implication, where n stands for an unknown integer.“If n is divisible by 10, then n is divisible by 5.”  a. Write the equivalent “or” form of this implication as an English sentence.  b. Write the logical negation of this implication as an English sentence. (Do not use “It isnot the case that…”)

Which of the following is the contrapositive of the sentence " If n²+4n+5 is odd, then n is even"?

Alice and Belle are perfect logicians. Each of them is given a different fraction, of the form , where n is a positive integer. They only know their own number and they know it's a different fraction from the other. They need to tell whose number is larger. The following conversation occurs: Alice: "I don't know". Belle: "I don't know either." Alice: "I still don't know." Belle: “Now I know whose number is larger." Alice: "Now I know your number too." Who has the larger number? What is the difference of the denominators? (A) Alice. 1. (B) Alice. 2. (C) Belle. 1. (D) Belle. 2.

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