Math 330 - 01 Homework (Spring 2022)
Problem Set 13 (complete) Due: 05/09/2022
Let f:A→B and g:C→D be functions. Define f×g:A×C→B×D by (f×g)(a,c)=(f(a),g(c)).
Prove that if f and g are surjective, then so is f×g.
Prove that the function f:Z→N given by f(m)={2mif m>0, −2m+1if m≤0, is bijective.
Prove that if A and B are finite sets, then so is A∪B. Morevoer, if A and B are disjoint, then |A∪B|=|A|+|B|.
Prove Theorem 13.28. Hint: consider the function tan(x) from calculus.
Problem Set 12 (complete) Due: 05/02/2022. Board presentation: 05/06/2022
Prove the converse of Prop 11.2
Prove that for all x,y,z,w∈R with z≠0≠w, xz+yw=xw+yzzwandxzyw=xyzw
Consider the set A={x∈Q∣x2<2} Show that A is non-empty and has an upper bound in Q, but does not have a least upper bound in Q. Hint: by way of contradiction, assume A has a least upper bound u in Q, and compare it with √2.
Prove Prop. 11.21.iii
Problem Set 11 (complete) Due: 04/19/2022. Board presentation: 04/22/2022
Prove part (iv) of lemma stated in class:
for x∈R and r∈R+,
(iv) |x|≤r iff x≤r and −x≤r.
(Hint: use part (iii) of the same lemma.
Prove Prop. 10.10.iii (Hint: use 10.8.iv)
Prove Prop. 10.13.ii
Prove Prop. 10.17 (Hint: use induction)
Problem Set 10 (complete) Due: 04/11/2022. Board presentation: 04/15/2022
Let f:A→B and g:B→C be functions.
Prove Prop. 9.7.ii
Prove that if g∘f is surjective, then g is surjective.
Prove Prop. 9.10.ii
Prove Prop. 9.15 (Hint: induction)
Prove Prop. 9.18
Previous Homework
Home