This chapter builds the bridge between two mathematical ideas: a relation, which describes a connection between elements of two sets, and a function, which is a special type of relation with a unique assignment rule.
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Cartesian Product
The Cartesian product of two sets A and B, written A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
If |A| = m and |B| = n, then |A × B| = m × n.
Note: A × B ≠ B × A (ordered pairs matter).
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Relations
A relation R from set A to set B is any subset of A × B. We write a R b to mean (a, b) ∈ R.
- Domain of R: Set of all first elements of ordered pairs in R.
- Range of R: Set of all second elements of ordered pairs in R.
- Codomain: The full set B (may be larger than the range).
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Functions
A function f from A to B (written f : A → B) is a relation where every element of A is paired with exactly one element of B.
- A is the domain, B is the codomain, and the set of actual output values is the range.
- Types of functions:
- One-one (Injective): Different inputs give different outputs (a1 ≠ a2 ⇒ f(a1) ≠ f(a2)).
- Onto (Surjective): Every element of B is the image of at least one element of A (Range = Codomain).
- One-one and onto (Bijective): Both injective and surjective.
- Some real-valued functions:
- Identity function: f(x) = x
- Constant function: f(x) = c
- Modulus function: f(x) = |x|
- Signum function: f(x) = 1 if x > 0; 0 if x = 0; -1 if x < 0
- Greatest integer function: f(x) = [x] (largest integer ≤ x)
Algebra of functions: If f and g are functions with domain D, then (f + g)(x) = f(x) + g(x), (f − g)(x) = f(x) − g(x), (f·g)(x) = f(x)·g(x), and (f/g)(x) = f(x)/g(x) provided g(x) ≠ 0.
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Worked Examples
If A = {1, 2} and B = {a, b, c}, write A × B.
A × B = {(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)}
Let A = {1,2,3} and R = {(a,b) : b = a + 1, a,b ∈ A}. List R, domain, and range.
R = {(1,2),(2,3)}; Domain = {1,2}; Range = {2,3}
Is the relation f = {(1,2),(2,4),(3,6),(4,8)} a function from A = {1,2,3,4} to B = {2,4,6,8,10}?
Yes. Every element of A has exactly one image in B.
Determine the domain of f(x) = 1/(x − 3).
f(x) is undefined when x = 3. Domain = R − {3}.
Find the range of f(x) = x2.
x2 ≥ 0 for all real x. Range = [0, ∞).
If f(x) = 2x + 3, find f(0), f(1), f(-1).
f(0) = 3, f(1) = 5, f(-1) = 1.
Show that f(x) = 2x + 1 (f: R → R) is one-one and onto.
One-one: f(a) = f(b) ⇒ 2a+1 = 2b+1 ⇒ a = b. Onto: for any y ∈ R, x = (y-1)/2 gives f(x) = y. So f is bijective.
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Common mistakes
- Confusing relation with function: a function needs every domain element paired with exactly one co-domain element.
- Confusing range and codomain — the range is always a subset of the codomain, not necessarily equal.
- When finding domains, forgetting to exclude values that cause division by zero or negative square roots.
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Summary
Cartesian products generate ordered pairs; relations are subsets of these products; functions are special relations with unique assignment. Key properties (one-one, onto) describe how domain maps to codomain.