A set is a collection of objects. Sets can be denoted in two different ways, for example the set $S$ containing two numbers $1$ and $-1$ can be denoted by $$S = \{ -1,1\} = \{ x : x^2 = 1 \text{ and } x \in \mathbb{R}\}$$ which reads like $S$ is the set containing $1$ and $-1$, and $S$ is the set containing $x$ where $x^2$ is equal to $1$ and $x$ is a real number.
If every element of a set $X$ is contained within a set $Y$, then we say that $X$ is a subset of $Y$, and we denote this by $X \subset Y$.
The empty set, denoted by $\emptyset$, denotes the set containing no objects.
The intersection of $X$ and $Y$, denoted by $X \cap Y$, is the set containing all objects that are in both $X$ and $Y$. In other words: $$X \cap Y = \{ x : x \in X \text{ and } x \in Y \}$$ The intersection of multiple sets is usually denoted in the following way: $$ \bigcap_{i \in I} X_i = \{x : \text{ for each } i \in I, x \in X_i\} $$
The union of $X$ and $Y$, denoted by $X \cup Y$, is the set containing all objects that are in either $X$ or $Y$. In other words: $$X \cup Y = \{x : x \in X \text{ or } x \in Y\}$$ The union of multiple sets is usually denoted in the following way: $$ \bigcup_{i \in I} X_i = \{x : x \in X_i \text{ for some } i \in I\}$$
If $X \subset S$, then the complement of $X$ in $S$ is the set of all elements of $S$ which are not elements of $X$. In other words: $$X^\complement = \{ x \in S : x \notin X\}$$
Two sets are said to be disjoint if they have no element in common. Similarly, a collection of sets is said to be disjoint if no two sets have an element in common.
An ordered pair $(a,b)$ is an object in which the order of $a$ and $b$ matter.
Given two sets $X$ and $Y$, we define the cartesian product of $X$ and $Y$, denoted by $X \times Y$ to be $$X \times Y = \{(x,y) : x \in X, y \in Y\}$$
If $X$ and $Y$ are sets, a function from $X$ to $Y$ is a rule that associates with each element of $X$ a definite element of $Y$. The statement "$f \text{ is a function from } X \text{ to } Y$" is often written $f : X \rightarrow Y$
If $f : X \rightarrow Y$ and $g : Y \rightarrow Z$, then we can define the composite function as $$ (g \circ f)(x) = g(f(x))$$
A function $f: X \rightarrow Y$ is called one-to-one if $f(x_1) = f(x_2)$ only if $x_1 = x_2$. This is also called being injective.
A function $f: X \rightarrow Y$ is called onto if each element of $Y$ corresponds to some value of $X$ under $f$, or in other words if for every $y \in Y$, there exists some $x \in X$ such that $f(x) = y$. This is also called being surjective.
A function that is both one-one and onto is called one-one onto, or bijective.
A bijective function $f$ has a corresponding inverse function $f^{-1}$, such that $f^{-1}(y) = x$ if and only if $f(x) = y$.
If $f:X \rightarrow Y$ a funtion with $X' \subset X$, then the subset of $Y$ given by $$f(X') = \{f(x) : x \in X\}$$ is called the image of $X'$. Similarly, if $Y' \subset Y$, then the subset of $X$ given by $$f^{-1}(Y') = \{x \in X: f(x) \in Y'\}$$ is called the inverse image of $Y'$. In layman terms, the image of a set is what that set maps to, the inverse image of a set is everything that maps to that set.