Operations On Sets Review

# Operations on Sets Review

We will now review some of the recent material regarding operations on sets.

- On the
**Unary and Binary Operations on Sets**page we defined a**Unary Operation**on a set $S$ to be a bijection function $f : S \to S$. The simplest unary operation is the negation operation $- : \mathbb{R} \to \mathbb{R}$ defined for each real number $x \in \mathbb{R}$ by $-(x) = -x$.

- We then defined a
**Binary Operation**on a set $S$ to be a function $f : S \times S \to S$ such that for each $(x, y) \in S \times S$ we have that $f(x, y)$ is defined, $f(x, y) \in S$, and $f(x, y)$ is unique. The simplest binary operation is the addition operation $+ : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ defined for all $x, y \in \mathbb{R}$ to be $x + y$.

- On the
**Functions that are Not Binary Operations**page we looked at an example of a function that is not a binary operation.

- We then began to look at properties of binary operations.

- On the
**Associativity and Commutativity of Binary Operations**we said that a binary operation $*$ defined on a set $S$ is**Associative**if for all $a, b, c \in S$ we have that:

\begin{align} \quad a * (b * c) = (a * b) * c \end{align}

- We said that $*$ is
**Commutative**if for all $a, b \in S$ we have that:

\begin{align} \quad a * b = b * a \end{align}

- On the
**Distributivity of Binary Operations**we said that if $*$ and $+$ are binary operations defined on a set $S$ then $*$ is**Left Distributive**if for all $a, b, c \in S$ we have that:

\begin{align} \quad a * (b + c) = (a * b) + (a * c) \end{align}

- And $*$ is
**Right Distributive**if for all $a, b, c \in S$ we have that:

\begin{align} \quad (a + b) * c = (a * c) + (b * c) \end{align}

- We said that $*$ is
**Distributive**if it is both left distributive and right distributive.

- On the
**Identity and Inverse Elements of Binary Operations**page we said that if $S$ is a set and $*$ is a binary operation on $S$ then an element $e \in S$ is an**Identity Element**if for all $a \in S$ we have that:

\begin{align} \quad a * e = a \quad \mathrm{and} \quad e * a = a \end{align}

- We proved that if such an identity element exists then it must be unique.

- Furthermore, if an identity element $e$ exists and $a, b \in S$ are such that $a * b = e$ and $b * a = e$ then $b$ is said to be the
**Inverse**of $a$ which we denote $a^{-1} = b$.