$\boxed{a \in X}$ means $a$, as a value, is an element of set $X$. This is a boolean operator that can be evaluated to be true or false.
$a = b$ means that the value of $a$ and the value of $b$ are the same, hence interchangeable.
As a predicate, $P(a)$ is a function that returns a boolean (true/false) value, $a$ is the parameter, and $P$ is the name of the predicate. This is a useful notation that takes the place of a potentially complicated expression. The actual definition of $P(a)$ depends on the context.
$\boxed{\forall e}(P(e))$ means "for every value $e$, $P(e)$ is true." This universally-quantified expression returns a boolean value.
$\boxed{\exists e}(P(e))$ means "there is at least one value $e$ such that $P(e)$ is true." This existentially-quantified expression returns a boolean value.
$\forall e((e \in (\boxed{X \cap Y)}) \Leftrightarrow ((e \in X) \wedge (e \in Y)))$ is definitive, $X \cap Y$ is the intersection of the sets $X$ and $Y$. The term "definitive" means an expression true by definition.
$\forall e((e \in (\boxed{X \cup Y)}) \Leftrightarrow ((e \in X) \vee (e \in Y)))$ is definitive. $X \cup Y$ is the union of the sets $X$ and $Y$.
$\forall e((e \in (\boxed{X - Y})) \Leftrightarrow ((e \in X) \wedge \neg(e \in Y)))$ is definitive. $X - Y$ is the difference of the sets $X$ and $Y$.
$\boxed{X \subseteq Y} \Leftrightarrow \forall e(((e \in X) \Rightarrow (e \in Y)))$ is definitive. $X \subseteq Y$ evaluates whether $X$ is a subset of $Y$, it says "every element $e$ in set $X$ is also in set $Y$."
$\boxed{X \subset Y} \Leftrightarrow ((X \subseteq Y) \wedge (\exists e((e \in Y) \wedge \neg(e \in X))))$ is definitive. $X \subset Y$ evaluates whether $X$ is a proper subset of $Y$. There needs to be one element in $Y$ but not in $X$.
$\forall e((e \in \boxed{\{e|P(e)\}}) \Leftrightarrow (P(e)))$ is definitive. This is a notation to describe the membership of a set with an infinite number of elements where $P$ is a predicate.
To describe all the members of a set that has a finite number of elements:
The general BNF syntax is as follows: { [ element { ,element }] }*
Anything that is bold is a terminal; it should be a part of the expression verbatim.
Anything that is italic is a token; it is a placeholder of something else.
Anything that is between brackets [] is optional; there may be zero or one occurrence.
Anything that is between braces {} is a group.
Anything that is followed immediately by an asterisk * can occur any number of times, including zero.
Given the syntax described, the ordering of elements in this notation is not important. As an example, $\{a,b,c\} = \{a,c,b\} = \{b,a,c\} = \{b,c,a\} = \{c,a,b\} = \{c,b,a\}$.
$\boxed{\{\}}$ is known as the empty set.
$\boxed{|X|}$ is the cardinality of the set $X$, the cardinality of a set is similar to the number of elements in the set.
$\boxed{\forall e \in X(P(e))}$ is an abbreviation of $\forall e((e \in X) \Rightarrow (P(e)))$, it says "for every element $e$ in set $X$, $P(e)$ is true."
$\boxed{\exists e \in X(P(e))}$ is an abbreviation of $\exists e((e \in X) \wedge (P(e)))$, it says "there is at least one element $e$ in set $X$ such that $P(e)$ is true."
The general BNF of a tuple is as follows: ( [ element { ,element }] )*. The ordering of values in a tuple is significant.
$\forall e(\forall f(((e,f) \in \boxed{X \times Y}) \Leftrightarrow ((e \in X) \wedge (f \in Y))))$ is definitive. $X \times Y$ is called the cartesian product of the sets $X$ and $Y$, each element in $X \times Y$ is a two-tuple (tuple with two items), where the first item in the 2-tuple is an element from $X$ and the second item is an element from $Y$.
Given $X$ and $Y$ are sets, then $(\boxed{X=Y}) \Leftrightarrow ((X \subseteq Y) \wedge (Y \subseteq X))$, this defines the equality of two sets, and it lays the foundation of understanding nested sets. This definition is also equivalent to $(X=Y) \Leftrightarrow (\forall e((e \in X) \Leftrightarrow (e \in Y)))$.