Algebra/Chapter 2/Logic and Proofs

2.6: Logic and Proofs

Practice Problems
 Problem 2.80 (Using Properties of Numbers) Justify each step, using the properties of communativity and associativity in proving the following identities.

$$a.\ (a + b) + (c + d) = (a + d) + (b + c)$$ $$b.\ (a + b) + (c + d) = (a + c) + (b + d)$$ $$c.\ (a - b) + (c - d) = (a + c) + (-b - c)$$ $$d.\ (a - b) + (c - d) = (a + d) - (b + c)$$ $$e.\ (a - b) + (c - d) = (a - d) + (c - b)$$ $$f.\ (a - b) + (c - d) = -(b + d) + (a + c)$$ $$g.\ ((a + b) + c) + d = (a + c) + (b + d)$$ $$h.\ (a - b) - (c - d) = (a - c) + (d - b)$$

 Problem 2.81 (Using Properties of Numbers) Determine if the following statements are true or false. Justify your conclusions.

a. If $$a$$, $$b$$, and $$c$$ are integers, then the number $$ab + bc$$ is an even number. b. If $$b$$ and $$c$$ are odd integers, and $$a$$ is an integer, then the number $$ab + bc$$ is an even number.

 Problem 2.82 (Using Properties of Numbers) We define an integer $$a$$ to be of


 * Type I if $$a=4n$$ for some integer $$n$$
 * Type II if $$a=4n + 1$$ for some integer $$n$$
 * Type III if $$a=4n + 2$$ for some integer $$n$$
 * Type IV if $$a=4n + 3$$ for some integer $$n$$

a. Provide at least two examples of each of the four types of integers above. b. Is it true that if $$a$$ is even, then it is of type I or III? Justify your answer. c. Is it true that if $$a*b$$ is of type I, whenever $$a$$ or $$b$$ are of type III? Justify your answer.

 Problem 2.83 (Using Properties of Numbers) For all real numbers $$x$$ and positive integers $$n$$, show that:

$$(1-x)(1+x+x^2+...+x^{n-1}+x^n)=1-x^{n+1}$$