Precalculus - Assignment 1

Submit by: September 12

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1. Let $f:A\to B$ and $g:B\to A$ such that $g\circ f =\mathrm{id}_A$. State whether the following conclusions are valid. For valid conclusions give a proof. For invalid conclusions give a counterexample.
   
   1. $f$ is a bijection.
   2. $f$ is onto.
   3. $f$ is one-one.
   4. $g$ is a bijection.
   5. $g$ is onto.
   6. $g$ is one-one.
2. Let $A$ and $B$ be sets with 3 and 4 elements respectively. Can there be a bijection between these sets?
   
3. Exhibit a bijection from the set of even natural numbers $2\mathbb N$ to the set of all natural numbers $\mathbb N$.
   
4. Exhibit a bijection from $\mathbb N$ to the set of integers $\mathbb Z$.
   
5. Are the following functions one-one or onto?
   
   1. $f:[0,1]\to[a,b]$, $f(x)=bx+(1-x)a$.
   2. $f:\mathbb R\to\mathbb R$, $f(x)=x^2+x+1$.
   3. $f:\mathbb R\to\mathbb R$, $f(x)=x+|x|$.
6. Let $f:\mathbb N\to A$ and $g:\mathbb N\to B$ be surjective. Show there is a surjective map $h:\mathbb N\to A\cup B$.
   
7. Let $f:\mathbb R^2\to\mathbb R$ be defined by $f(x,y)=xy$. What are $f^{-1}(r)$ for $r\in\mathbb R$ and $f^{-1}([a,b])$? Draw pictures of these inverse images.
   
8. Let $f:X\to Y$. Show that
   
   1. $f$ is onto iff $f(f^{-1}(B))=B$ for every $B\subset Y$.
   2. $f$ is one-one iff $f^{-1}(f(A))=A$ for every $A\subset X$.