# Group Theory 2: Preliminaries (Functions)

In my previous notes here, I mentioned some about logical symbols. The logical symbols I will use often are $\forall$ which means “for all”, “for any”, “for each”, or “for every” depending on the context, $\exists$ which means “there exists”, and $\ni$ which means “such that” (don’t be confused with $\in$ which means “be an element of”). We also use s.t. for “such that.” There are also $\Longrightarrow$ which means “implies” and $\Longleftrightarrow$ which means “if and only if.” I guess these pretty much cover what we use most of time.

Now lets review about functions in a more formal way. Let $X$ and $Y$ be two non-empty sets. The the Cartesian product $X\times Y$ of $X$ and $Y$ is defined as the set
$$X\times Y=\{(x,y): x\in X,\ y\in Y\}.$$
A subset $f$ of the Cartesian product $X\times Y$ (we write $f\subset X\times Y$) is called a graph from $X$ to $Y$. A graph $f\subset X\times Y$ is called a function from $X$ to $Y$ (we write $f: X\longrightarrow Y$) if whenever $(x,y_1),(x,y_2)\in f$, $y_1=y_2$. If $f: X\longrightarrow Y$ and $(x,y)\in f$, we also write $y=f(x)$. A function $f: X\longrightarrow Y$ is said to be one-to-one or injective if whenever $(x_1,y),(x_2,y)\in f$, $x_1=x_2$. This is equivalent to saying $f(x_1)=f(x_2)$ implies $x_1=x_2$. A function $f: X\longrightarrow Y$ is said to be onto or surjective if $\forall y\in Y$ $\exists x\in X$ s.t. $(x,y)\in f$. A function $f: X\longrightarrow Y$ is said to be one-to-one and onto (or bijective) if it is both one-to-one and onto (or both injective and surjective).

Let $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ be two functions. Then the composition or the composite function $g\circ f: X\longrightarrow Z$ is defined by $g\circ f(x)=g(f(x))$ $\forall x\in X$. The function composition $\circ$ may be considered as an operation and it is associative.

Lemma. If $h: X\longleftrightarrow Y$, $g:Y\longleftrightarrow Z$ and $f:Z\longleftrightarrow W$, then $f\circ(g\circ h)=(f\circ g)\circ h$.

Note that $\circ$ is not commutative i.e. it is not necessarily true that $f\circ g=g\circ f$ even when both $f\circ g$ and $g\circ f$ are defined.

The following lemmas will be useful when we study group theory later.

Lemma. If both $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ are one-to-one, so is $g\circ f: X\longrightarrow Z$.

Lemma. If both $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ are onto, so is $g\circ f: X\longrightarrow Z$.

As an immediate consequence of combining these two lemmas, we obtain

Lemma. If both $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ are bijective, so is $g\circ f: X\longrightarrow Z$.

If $f\subset X\times Y$, then the inverse graph $f^{-1}\subset Y\times X$ is defined by
$$f^{-1}=\{(y,x)\in Y\times X: (x,y)\in f\}.$$
If $f: X\longrightarrow Y$ is one-to-one and onto (bijective) then its inverse graph $f^{-1}$ is a function $f^{-1}: Y\longrightarrow X$. The inverse $f^{-1}$ is also one-to-one and onto.

Lemma. If $f: X\longrightarrow Y$ is a bijection, then $f\circ f^{-1}=\imath_Y$ and $f^{-1}\circ f=\imath_X$, where $\imath_X$ and $\imath_Y$ are the identity mappings of $X$ and $Y$, respectively.

Let $A(X)$ be the set of all one-to-one functions of $X$ onto $X$ itself. Then $(A(X),\circ)$ is a group. If $X$ is a finite set of $n$-elements (we may conveniently say $X=\{1,2,\cdots,n\})$, then $(A(X),\circ)$ is a finite group of order $n!$, called the symmetric group of degree $n$. The symmetric group of degree $n$ is denoted by $S_n$ and the elements of $S_n$ are called permutations.