Homework Assignment 3
Due: Wednesday, September 27 1. Let f : X Y . (a) Show that f is surjective if and only if f f 1 [B ] = B for all subsets B Y . (b) Show that f is injective if and only if f 1 [f (A)] = A for all subsets A X . 2. Provide a proof or a
Homework Assignment 6
Due: Wednesday, November 22 1. Prove that any two consecutive Fibonacci numbers are relatively prime. 2. Find the smallest positive integer a for which the equation 1602 x + 1170 y = 106 + a has a solution in integers x and y . Also,
Homework Assignment 7
Due: Friday, December 8 1. (a) Find all integers x that satisfy 374x 204 (b) Solve the system x 13 (mod 78) x 313 (mod 350). (c) Solve the linear congruence system 115x 41 (mod 78) 144x 141 (mod 350). 2. Exercise 5.4(a) in the lectur
Solutions to Homework Assignment 2 Math 74, Fall 2006
October 1, 2006
1. (a) If P and Q are statements, then using De Morgans laws we see that [P Q] P Q P Q. Likewise, [P Q] P Q, and ( [P Q]) ( [Q P ]) (P Q). So we see that we can build all of the logical
Solutions to Homework Assignment 3 Math 74, Fall 2006
October 1, 2006
1. (a) Proposition 1. Let f : X Y . Then f is surjective if and only if f f 1 [B ] = B for every subset B Y . Proof. First, assume that f is surjective. Choose an arbitrary subset B Y .
Solutions to Homework Assignment 4 Math 74, Fall 2006
October 6, 2006
1. Denition 1. A natural number n is a perfect square if there exists an integer k such that n = k 2 . Theorem 2. If n N then n Q if and only if n is a perfect square. Proof. Clearly if
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