I have search out my whole maths book for it..but didn't got this question in my book and I need to learn it for my exam..So if you people can please solve this for me?
$S=\{0,1,2,3\}$. Show that $(S,+)$ is a Semi group where $+$ defines addition Modulo 4?
Waiting for your reply please..
In short a semigroup is a set $S$ together with some binary operation $\star$ on it that is associative. That means that for every pair of elements $x,y\in S$ there is an element $x\star y\in S$ and the rule $\left(x\star y\right)\star z=x\star\left(y\star z\right)$ is obeyed. An example is $\mathbb{N}=\left\{ 1,2,\dots\right\} $ together with the addition as operation, denoted by $+$. It is a semigroup since rule $\left(n+m\right)+k=n+\left(m+k\right)$ is satisfied. In your case we deal with $S=\left\{ 0,1,2,3\right\} $ and the operation is addition modulo $4$. To show that it is a semigroup it must be verified that the rule of associativity is valid in this situation. That is what you are asked to do.
Caution: check whether this definition of semigroup agrees with the one in your book. It can be that it is also demanded that $S$ contains some specific element $e$ having the property that $e\star s=s\star e=s$ for each $s\in S$. I would speak of a 'monoid' instead of a semigroup, but others don't. If that is the case then the mentioned example $\mathbb{N}=\left\{ 1,2,\dots\right\} $ with addition should be replaced by $\mathbb{N}\cup\{0\}$ and $0$ is the specific element (called identity or unit).