maths

MATH 127 Homework Assignment on 1.3-1.5 Due on September 26, 2017

1. Determine whether each pair is logically equivalent. Show that your answer is correct. a) P Q and (P Q) b) (P Q) R and ((P Q) R) c) (P) (P Q) and (Q P)

2. Textbook section 1.3 # 13

3. Form the i) converse, ii) contrapositive, iii) inverse of each statement. a) If a function has a constant derivative, then the function is linear. b) If , then or . c) If is a rational number, then is a rational number. d) For a matrix to be invertible, it is necessary that the matrix have a nonzero determinant.

4. Form the negation of each statement. (You may not simply insert the word “not”.) a) If a function is a polynomial, then the function is rational. b) The number 8 is both even and a power of 2. c) It is the weekend or I’m going to school. d) The number equals , but the number does not.

e) The number is a multiple of if and only if divides .

5. For each of the following a) - g), i) Determine whether each statement is universal or existential. ii) Form the negation of each statement.

a) For all real numbers , b) For all real numbers , there is a natural number such that c) There exists a real number such that for all real numbers , d) Every time it rains there is a rainbow in the sky. e) There is at least one passenger for every car on the road.