Unit 2: Relations and Functions

1. Cartesian Product of Sets
2. Relations: Domain and Range
3. Functions and Their Representation
4. Graphs of Special Functions
5. One-One and Onto Functions
6. Exam Focus Enhancements

1. Cartesian Product of Sets

The Cartesian Product of two non-empty sets A and B is the set of all ordered pairs (a, b) such that a belongs to A and b belongs to B.

Notation: A × B = { (a, b) : a ∈ A and b ∈ B }

If set A has p elements and set B has q elements, then A × B has p × q elements.
Order matters: (a, b) ≠ (b, a) unless a = b.

2. Relations: Domain and Range

A Relation (R) from set A to set B is a subset of the Cartesian product A × B. It is derived by describing a relationship between the first element and the second element of the ordered pairs.

Key Terms:

Domain: The set of all first elements of the ordered pairs in a relation R.
Range: The set of all second elements of the ordered pairs in a relation R.
Co-domain: The entire set B in a relation from A to B. (Note: Range ⊆ Co-domain).

3. Functions and Their Representation

A Function (f) from set A to set B is a special type of relation where every element of set A has one and only one image in set B.

Conditions for a function:

Every element in the domain (Set A) must be associated with an element in the co-domain.
No element in the domain can have more than one image.

4. Graphs of Special Functions

Visualizing functions through graphs is crucial for understanding their behavior:

Function Name	Expression	Graphical Behavior
Identity Function	f(x) = x	A straight line passing through the origin at 45°.
Modulus Function	f(x) = \|x\|	A V-shaped graph with the vertex at (0,0).
Greatest Integer Function	f(x) = [x]	A "step" or "staircase" graph. It rounds down to the nearest integer.
Signum Function	f(x) = x/\|x\|	Returns 1 for x > 0, -1 for x < 0, and 0 for x = 0.

5. One-One and Onto Functions

These classifications describe how elements are mapped between sets:

One-One (Injective) Function

A function is One-One if distinct elements in the domain have distinct images in the co-domain.
Mathematically: if f(x₁) = f(x₂), then x₁ = x₂.

Onto (Surjective) Function

A function is Onto if every element in the co-domain is the image of at least one element in the domain.
Mathematically: Range = Co-domain.

Bijective Function

A function that is both One-One and Onto is called Bijective. It represents a perfect "one-to-one correspondence."

6. Exam Focus Enhancements

Exam Tips

Vertical Line Test: To check if a graph represents a function, draw a vertical line. If it touches the graph more than once, it is NOT a function.
Range vs Co-domain: Always double-check if every element in B is used. If yes, it's Onto. If no, it's Into.
Modulus Steps: When graphing |x|, remember it always returns non-negative values. The graph will never go below the x-axis.

Common Mistakes

Confusing Relation with Function. Every function is a relation, but every relation is not necessarily a function.
In [x] (Greatest Integer Function), students often round to the "nearest" integer. Remember: it always rounds down (e.g., [-1.1] = -2).
Forgeting the Codomain. Range is what comes out of the function; the Codomain is the target set provided in the definition.

Frequently Asked Questions

Q: Can a function have an empty domain?
A: Technically, an empty function exists from an empty set, but for practical exam purposes, domains are non-empty.

Q: What is a Many-One function?
A: A function where two or more different elements in the domain have the same image in the co-domain (e.g., f(x) = x²).