MAT301 ASSIGNMENT 1 DUE DATE: WEDNESDAY MAY 18, 2016 AT THE BEGINNING OF YOUR TUTORIAL Question 1. Define the following relation on the integers: a b 2|(a b) Prove that this is an equivalence relation. How many distinct equivalence classes are there? List them. Question 2. Let A B. Let |A| and |B| be the number of elements in A and B, respectively. (a) Suppose |A| = |B| < (that is, both sets are finite sets). Prove that A = B. (b) Give an example showing that part (a) is false if the sizes of the sets are infinite. That is, give an example of two sets A and B with A B, |A| = |B| = , but A 6= B. Question 3. Suppose f : A B is a function with |A| = |B| = n < (that is, both sets are finite and are of the same size). Prove that f is injective (or 1-1) if and only if it is surjective (or onto). Question 4. Let \u001a\u0012 G= a a a a \u0013 \u001b : a R, a 6= 0 Prove that G is a group under (normal) matrix multiplication. Question 5. Let G = {x Q : 0 x < 1}. Define the following operation on G: ( a+b if 0 a + b < 1 ab= a + b 1 if a + b 1 Prove that (G, ) is an Abelian group. Question 6. (a) Let \u001a\u0012 \u0013 \u001b a b 2 G= : a, b, c R, ac 6= b b c Is G a group under (normal) matrix multiplication (either prove it is, or give an example showing one of the axioms fails)? (b) Let \u001a\u0012 G= a b c a \u0013 \u001b : a, b, c R, a = 6 bc 2 Is G a group under (normal) matrix multiplication (either prove it is, or give an example showing one of the axioms fails)? (c) Let \u001a\u0012 \u0013 \u001b a b G= : a, b, c, d Z, ad 6= bc c d Is G a group under (normal) matrix multiplication (either prove it is, or give an example showing one of the axioms fails)? Question 7. Suppose G is a group with the property that g 2 = e for all g G. Prove that G is Abelian. 1 MAT301 ASSIGNMENT 1 DUE DATE: WEDNESDAY MAY 18, 2016 AT THE BEGINNING OF YOUR TUTORIAL Question 1. Define the following relation on the integers: a b 2|(a b) Prove that this is an equivalence relation. How many distinct equivalence classes are there? List them. Question 2. Let A B. Let |A| and |B| be the number of elements in A and B, respectively. (a) Suppose |A| = |B| < (that is, both sets are finite sets). Prove that A = B. (b) Give an example showing that part (a) is false if the sizes of the sets are infinite. That is, give an example of two sets A and B with A B, |A| = |B| = , but A 6= B. Question 3. Suppose f : A B is a function with |A| = |B| = n < (that is, both sets are finite and are of the same size). Prove that f is injective (or 1-1) if and only if it is surjective (or onto). Question 4. Let \u001a\u0012 G= a a a a \u0013 \u001b : a R, a 6= 0 Prove that G is a group under (normal) matrix multiplication. Question 5. Let G = {x Q : 0 x < 1}. Define the following operation on G: ( a+b if 0 a + b < 1 ab= a + b 1 if a + b 1 Prove that (G, ) is an Abelian group. Note: You do not have to show this operations is associative (you can assume it is). Question 6. (a) Let \u001a\u0012 \u0013 \u001b a b 2 G= : a, b, c R, ac 6= b b c Is G a group under (normal) matrix multiplication (either prove it is, or give an example showing one of the axioms fails)? (b) Let \u001a\u0012 G= a b c a \u0013 \u001b : a, b, c R, a = 6 bc 2 Is G a group under (normal) matrix multiplication (either prove it is, or give an example showing one of the axioms fails)? (c) Let \u001a\u0012 \u0013 \u001b a b G= : a, b, c, d Z, ad 6= bc c d Is G a group under (normal) matrix multiplication (either prove it is, or give an example showing one of the axioms fails)? Question 7. Suppose G is a group with the property that g 2 = e for all g G. Prove that G is Abelian. 1