1. ## Mathematics Sets

Help required1

Can anyone with help me with mathematics sets/
How can I determine the type of relationships in sets such as equivalence, inverse, reflexive, symmetrical and transitive?
here is ab example X = {1,2,3}

2. Sets in themselves do not have "equivalence" or "inverses", and they cannot be "reflexive", "symmetrical" or "transitive". We must take the set under a relation. For instance, let us examine the set you have given under the relation "=". Clearly, for $\displaystyle a,b,c \in X$ we have that $\displaystyle a=a$ (reflexive), if $\displaystyle a=b$ then $\displaystyle b=a$ (symmetric), and if $\displaystyle a=b$ and $\displaystyle b=c$ then $\displaystyle a=c$. Thus, "=" is called an equivalence relation (an absurdly simple example, I know).

I'm not entirely sure what you mean by inverses. If we take a set under an operation, for instance the set $\displaystyle \{0,1,2\}$ under addition modulo 3 then you have an identity element, $\displaystyle 0$ (an identity element is a neutral element - $\displaystyle a*id=id*a=a \text{ } \forall \text{ } a \in S$), and every element has an inverse (an element that will take the element back to the inverse): the inverse of 1 is 2 and the inverse of 2 is 1 as $\displaystyle 1+2=2+1=3 \equiv 0 \text{ mod } 3$.

Is that what you are looking for?