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 $x + y$.
• We then began to look at properties of binary operations.
(1)
\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:
(2)
\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:
(3)
\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:
(4)
\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.
(5)
\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 $a^{-1} = b$.