Basic concepts of set theory

If we denote a set using X, the expression \displaystyle x\in X means that the element x belongs to set X; so what \displaystyle x\notin X that does not belong

A common notation is to mark the elements of a set in braces, \displaystyle X=\{a,b,c\} or \displaystyle X=\{x| x\text{ meets a certain property}\} where the symbol is read as such that

Given two sets X, Y, their intersection \displaystyle X\cap Y and your union \displaystyle X\cup Y are two new sets defined by:

\displaystyle X\cap Y=\{a|a\in X\cap a\in Y\}
\displaystyle X\cup Y=\{a|a\in X\cup a\in Y\}

If all the elements of a set X are in another set Y, X is said to be subset of Y and denotes how \displaystyle X \subset Y\text{ or }X \subseteq Y; its negation is denoted as \displaystyle X \not\subset Y\text{ or }X \not\subseteq Y

If all the elements of a set X are equal to those of another set Y, which happens when \displaystyle X \subseteq Y\text{ and }Y \subseteq X, X is said to be equal to Y and denoted as X=Y; its negation is denoted as \displaystyle X \not\subseteq Y\text{ or }X \neq Y

When \displaystyle X \subset Y, the set of elements of Y that are not in X is denoted as \displaystyle Y \backslash X =\{a| a\in Y\cap a\not\in X\}

In some cases, the set Y is a kind of implicitly present total set, in those cases we have a set complementary of X, which has the same meaning as \displaystyle Y \backslash X

We could go further into set theory to achieve more rigor and depth, but there are two important concepts that we are going to emphasize: relationships and applications