Contrapositive and contradiction (4 points) Consider the following claim: For all integers m and n, if (m − n) is odd then m is odd or n is odd. (1) (2 points) Prove the claim using a proof by contrapositive (2) (2 points) Prove the claim using a proof by contradiction 2. Equivalence (4 points) Prove the following for all x ∈ R: x is rational ⇔ x − 5 is rational ⇔ x/3 is rational.

Suppose a pendulum is l meters long. the time, t,in seconds that it tales to swing back and forth once is given by t=2.01

Barbara deposited $300 in the bank. if the bank pays her simple interest at the rate of 4 percent per year, how much money will she have in her account at the end of 9 months?

Dexter read 11 science fiction books and 5 history books. lena read 9 science fiction books and 4 history books. compare the students’ ratios of science fiction books to history books. 1. determine the ratios’ form:    science fiction books to history books 2. write the ratios as fractions:    dexter: 115; lena: 94 3. rewrite with a common denominator. 4. compare. 5. answer the question. determine which shows the correct comparison of the ratios for this problem.