16. For each statement below find its truth value, write its negation and indicate the truth value of the negated statement. (a) (3r E N)(Vy E N)(r < y) (b) (Vr E N)(x2 > x) (c) (Vr Q)(Vy E Q)(r + y E Q) (d) (3r E N)(2r 3x +1 0) (e) (Vr e R)(2.r + 3r +1> 0) (f) (3z e Z)(2r + 3x +1 = 0) (g) ( E N)(212 +3r +1> 0) (h) (Vr E Z)(3y E Z)(y = 2x) 17. Give an informal proof: (a) if m is an odd integer, then m2 +1 is an even integer. (b) If m is an even integer and n is an odd integer, then mn is an even integer. (c) If mn is an odd integer, then m and n are both odd integers. (d) If m +n and mn are both even integers, then m and n are both even integers. (e) If m is an odd integer, then 4 is a factor of m2 - 1. (f) For an integer r, r+x2 +x is even if and only if r is even. (g) For every integer m, the integer m2 + 5m +7 is odd. (h) For every integer m, the integer 5m3 is odd if and only if 3m is an odd integer. (i) For integers m and n, the integer m2(n + 1) is even if and only if m is even or n is odd. (j) The natural number 110 cannot be written as the sum of two even natural numbers and an odd natural number. (k) There do not exist integers a and y such that 3x + 6y = 4. (1) There exist integers r and y such that 3r+ 5y = 1. (m) There exists a unique real number r such that for every real number y, ry+x-4 4y. (n) Every even integer is the sum of two odd integers. (o) The sum of four consecutive integers is an even integer. (p) For every integer r, the integer r2 +r is even. (g) For all odd integers m and n, the mumber 2 divides m2 + 3n. 18. Prove or disprove each conjecture: (a) For natural numbers a, b, and c, if a divides b and b divides a, then a = b. (b) The sum of three consecutive integers is an odd integer. (c) The sum of three consecutive integers is an even integer. (d) If x+ y is an irrational number, then both r and y are irrational. (e) the product of arational number and an irrational is an irrational number. 3