Math 330 - Spring 2015

Previous Homework

Homework Problems
 

Problems to hand in

Problems to presented on the board
 
Problem Set 7, Friday 11/06/15 (complete list)
  1. Prove Prop. 8.6
  2. Prove Thm. 8.43
  3. Let f:A → B and g:B → C be functions. Prove that if g ∘ f is injective then f is injective. Prove that if g ∘ f is surjective then g is injective.
  4. Give an example of a function with non-empty domain that is injective, but not surjective. Show that it has more than one left inverse.
  5. Give an example of a surjective function that is not injective. Show that it has more than one right inverse.
Friday 11/13/15 (complete list)
  1. Prove Prop. 8.6
  2. Prove Thm. 8.43
  3. Let f:A → B and g:B → C be functions. Prove that if g ∘ f is injective then f is injective. Prove that if g ∘ f is surjective then g is injective.
  4. Give an example of a function with non-empty domain that is injective, but not surjective. Show that it has more than one left inverse.
  5. Give an example of a surjective function that is not injective. Show that it has more than one right inverse.
Problem Set 6, Wednesday 10/28/15 (complete list)
  1. Let fn denote the n-th Fibonacci number. Prove by induction that: Σj=1n  f2j = f2n+1-1.
  2. Find all the partitions on as 4-element set A={a,b,c,d}. How many equivalence relations are there on A?
  3. Prop. 6.25.i
  4. Thm. 6.35. Hint: use induction, the Binomial Theorem, and Prop. 6.34.
Monday 11/02/15 (complete list)
  1. Let fn denote the n-th Fibonacci number. Prove by induction that: Σj=1n  f2j = f2n+1-1.
  2. Find all the partitions on as 4-element set A={a,b,c,d}. How many equivalence relations are there on A?
  3. Prop. 6.25.i
  4. Thm. 6.35. Hint: use induction, the Binomial Theorem, and Prop. 6.34.
Problem Set 5, Wednesday 10/14/15 (complete list)
  1. Prove Prop. 4.30 by induction on m.
  2. Prove Prop. 4.32. Hint: use Prop. 4.30
  3. Prove associativity of union and intersection of sets.
  4. Show, by a counterexample, that set difference is not associative.
  5. Prove Prop. 5.20.ii
Friday 10/23/15 (complete list)
  1. Prove Prop. 4.30 by induction on m.
  2. Prove Prop. 4.32. Hint: use Prop. 4.30
  3. Prove associativity of union and intersection of sets.
  4. Show, by a counterexample, that set difference is not associative.
  5. Prove Prop. 5.20.ii
Problem Set 4, Monday 10/05/15 (complete list)
  1. Prove Prop. 2.37 (appendix)
  2. Prove Prop. 4.6.iii
  3. Prove Prop. 4.11.ii
  4. Do project 4.12
  5. Prove Prop. 4.16.ii
Friday 10/16/15 (complete list)
  1. Prove Prop. 2.37 (appendix)
  2. Prove Prop. 4.6.iii
  3. Prove Prop. 4.11.ii
  4. Do project 4.12
  5. Prove Prop. 4.16.ii
Problem Set 3, Monday 09/21/15 (complete list)
  1. Prove Prop. 2.21 (Hint: proof by contradiction)
  2. Prove Prop. 2.23
  3. Prove Prop. 2.37 (appendix)
  4. Prove Prop. 2.38 (appendix)
Friday 09/25/15 (complete list)
  1. Prove Prop. 2.21 (Hint: proof by contradiction)
  2. Prove Prop. 2.23
  3. Prove Prop. 2.38 (appendix)
Problem Set 2, Wednesday 09/16/15 (complete list)
  1. Prove Prop. 1.25
  2. Prove Prop. 1.27.i,iv
  3. Prove Prop. 2.7
  4. Prove transitiviy of "≤".
Monday 09/21/15 (complete list)
  1. Prove Prop. 1.25
  2. Prove Prop. 1.27.i,iv
  3. Prove Prop. 2.7
  4. Prove transitiviy of "≤".
Problem Set 1, Wednesday 09/09/15 (complete list)
  1. Prove Prop. 1.7
  2. Prove that 1 + 1 ≠ 1. (Hint: assume otherwise, and get a contradiction). Can you prove that 1 + 1 ≠ 0?
  3. Prove Prop. 1.11.iv
  4. Prove Prop. 1.14
Friday 09/11/15 (complete list)
  1. Prove Prop. 1.7
  2. Prove that 1 + 1 ≠ 1.
  3. Prove Prop. 1.11.iv
  4. Prove Prop. 1.14
 

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