# Tagged Questions

75 views

### how to show a function is bijection

I have taken two numbers $p$ and $r$ where $p,r\in A = \{0,1,\ldots,4i + 1\}$ where $i\geq 1$ and $q\in B = \{0,1,\ldots,n-1\}$. Let $X$ contains all elements obtained by cartesian product of $A$ and ...
73 views

### Is this proof correct? Injective function $f: A \rightarrow B \iff$ function $g: B \rightarrow A$ is surjective

I've begun a course in "Real Analysis" recently and I have this trivial exercise. Could someone check if my proof is correct? Proposition: There exists Injective function $f: A \rightarrow B \iff$ ...
32 views

63 views

### How to prove that $f:\mathbb{N}\rightarrow X$ where $f$ maps to an element in a set, is a bijection?

Let $X$ and $Y$ be disjoint finite sets, $|X|=n$ and $|Y|=m$, so that we have the following bijections: $f:\mathbb{N}_n \rightarrow X$ and $g:\mathbb{N}_m \rightarrow Y$ I need to prove that ...
31 views

### Help to prove $f$ is surjective $\Leftrightarrow \forall y \in Y, (X \times \{y\} \cap G_f ) \ne \emptyset$

Let $f:X \rightarrow Y$ be a function with graph $G_f \subseteq X \times Y$. Prove that $f$ is surjective if and only if $\forall y \in Y, (X \times \{y\} \cap G_f ) \ne \emptyset$ I have some ...
This is the statement: If $f$ and $g$ are functions, the composition $g\circ f$ is a function with $$D(g\circ f)=\{x\in D(f):f(x)\in D(g)\}$$ $$R(g\circ f)=\{g(f(x)):x\in D(g\circ f)\}$$ The ...