Key Points

An ordered pair consists of two elements in a specific order, written as $(a, b)$. Two ordered pairs $(a, b)$ and $(c, d)$ are equal if and only if their corresponding elements are equal, meaning $a = c$ and $b = d$.

The Cartesian product of two non-empty sets A and B, denoted by $A \times B$, is the set of all possible ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. Formally, $A \times B = \{(a, b) : a \in A, b \in B\}$.

If a set A has $p$ elements and a set B has $q$ elements, then their Cartesian product $A \times B$ will have $p \times q$ elements. This is written as $n(A \times B) = n(A) \times n(B) = pq$.

A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product $A \times B$. It is formed by ordered pairs that satisfy a specific relationship.

For a relation R from A to B, the domain is the set of all first elements of the ordered pairs. The entire set B is the codomain. The range is the set of all second elements (images) of the ordered pairs, and it is always a subset of the codomain.

If a set A has $p$ elements and a set B has $q$ elements, the total number of possible relations from A to B is $2^{pq}$, because this is the total number of possible subsets of $A \times B$.

A relation $f$ from a set A to a set B is defined as a function if every element of set A has exactly one image in set B. This means no two distinct ordered pairs in $f$ have the same first element.

A function $f$ from set A to set B is denoted by $f: A \rightarrow B$. If $(a, b)$ is an ordered pair in the function, we write $f(a) = b$, where $b$ is the image of $a$ and $a$ is the pre-image of $b$.

A function is called a real valued function if its range is a subset of the set of real numbers $\mathbf{R}$. If its domain is also a subset of $\mathbf{R}$, it is called a real function.

The function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x) = x$ is the identity function. Both its domain and range are the set of all real numbers $\mathbf{R}$.

The function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x) = c$, where $c$ is a constant, is a constant function. Its domain is $\mathbf{R}$ and its range is the single element set {c}.

The modulus function is defined as $f(x) = |x|$. It is $f(x) = x$ for $x \geq 0$ and $f(x) = -x$ for $x < 0$. Its domain is $\mathbf{R}$ and its range is the set of non-negative real numbers, $[0, \infty)$.

The signum function is defined as $f(x) = 1$ if $x > 0$, $f(x) = 0$ if $x = 0$, and $f(x) = -1$ if $x < 0$. The domain is $\mathbf{R}$ and the range is the set $\{-1, 0, 1\}$.

The function $f(x) = [x]$ is the greatest integer function, which returns the greatest integer less than or equal to $x$. For example, $[3.7] = 3$ and $[-2.3] = -3$. Its domain is $\mathbf{R}$ and its range is the set of integers $\mathbf{Z}$.

For two real functions $f$ and $g$: $(f+g)(x) = f(x) + g(x)$, $(f-g)(x) = f(x) - g(x)$, $(fg)(x) = f(x)g(x)$, and $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ where $g(x) \neq 0$.