Problem Set 13 (complete) Due: 05/09/2022

1. Let $f:A\to B$ and $g:C\to D$ be functions. Define $f\times g:A\times C \to B\times D$ by $(f\times g)(a,c)=(f(a),g(c))$.
   Prove that if $f$ and $g$ are surjective, then so is $f\times g$.
2. Prove that the function $\ f:\Z \to \N$ given by $$f(m) = \cases {2m &if $m>0,$ \cr -2m+1 &if $m\leq 0,$ \cr}$$ is bijective.
3. Prove that if $A$ and $B$ are finite sets, then so is $A\union B$. Morevoer, if $A$ and $B$ are disjoint, then $|A\union B|=|A|+|B|$.
4. 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

1. Prove the converse of Prop 11.2
2. Prove that for all $x,y,z,w\in\R$ with $z\neq 0\neq w$, $$\frac{x}{z}+\frac{y}{w}=\frac{xw+yz}{zw}\qquad\textrm{and}\qquad\frac{x}{z}\frac{y}{w}=\frac{xy}{zw}$$
3. Consider the set $$A=\{x\in\Q\mid x^2<2\}$$ 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 $\sqrt{2}$.
4. Prove Prop. 11.21.iii

Problem Set 11 (complete) Due: 04/19/2022. Board presentation: 04/22/2022

1. Prove part (iv) of lemma stated in class:
   for $x\in\R$ and $r\in\R^+$,
   (iv) $|x| \leq r$ iff $x \leq r$ and $-x \leq r$.
   (Hint: use part (iii) of the same lemma.
2. Prove Prop. 10.10.iii (Hint: use 10.8.iv)
3. Prove Prop. 10.13.ii
4. Prove Prop. 10.17 (Hint: use induction)

Problem Set 10 (complete) Due: 04/11/2022. Board presentation: 04/15/2022

1. Let $f:A\to B$ and $g:B\to C$ be functions.
   1. Prove Prop. 9.7.ii
   2. Prove that if $g\circ f$ is surjective, then $g$ is surjective.
2. Prove Prop. 9.10.ii
3. Prove Prop. 9.15 (Hint: induction)
4. Prove Prop. 9.18

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