Fundamentals of Functions and Relations

Introduction to Functions and Relations

Understanding the language of mathematics begins with the concepts of relations and functions. These ideas form the backbone of algebra, calculus, and many advanced topics. In this course we will explore how to count relations, test function properties such as injectivity and surjectivity, work with composites, solve equality problems, and interpret inverses and identity functions. Each section includes key takeaways, memory aids, and examples drawn from a quiz format.

Counting Relations Between Finite Sets

How many distinct relations exist from A to B?

Given A = {1,2,3} and B = {x,y}, a relation is any subset of the Cartesian product A × B. The product contains every ordered pair formed by taking one element from A and one from B.

• Size of A × B: |A|·|B| = 3·2 = 6.
• Each of the 6 pairs can be either included or excluded from a relation.
• Number of possible subsets = 2⁶ = 64.

How to Remember

• Mnemonic: “Pairs = product; choices = 2 to the power of pairs.”
• Tip: Count the pairs first, then raise 2 to that count.

Injectivity, Surjectivity, and Bijectivity

Detecting a Failure of Injectivity

Consider the function f: A → B defined by f(1)=2, f(2)=3, f(3)=2. A function is injective when distinct inputs produce distinct outputs.

• Here, 1 and 3 both map to 2, violating the one‑to‑one rule.
• Surjectivity (every element of B is hit) and the fact that it is a function are still satisfied.
• Therefore, injectivity fails.

How to Remember

• Mnemonic: “I” in Injectivity stands for “Individual outputs”.
• Tip: Find two different domain elements that share the same image; if you can, the function is not injective.

Injectivity vs. Surjectivity for a Real‑Valued Function

Let g(x) = e^{−x} + 1 for all real x. To classify its properties:

• Derivative g'(x) = −e^{−x} < 0 for every x, so g is strictly decreasing → injective.
• Since e^{−x} > 0, the range of g is the interval (1, ∞). Values ≤ 1 are never attained, so g is not surjective onto ℝ.
• Conclusion: g is injective but not surjective.

How to Remember

• Mnemonic: “Downward DE‑crease → One‑to‑One, but never hits the floor (≤1).”
• Tip: Use the sign of the derivative for monotonicity (injectivity) and examine limits to locate the actual output interval (surjectivity).

Composition of Functions and Injectivity

When Does a Composite Remain Injective?

For functions f: A → B and g: B → C, the composite h = g ∘ f maps A directly to C. The crucial question: which condition guarantees that h is injective?

• If the outer function g is injective, then distinct outputs of f cannot be collapsed by g. The inner function f may be any mapping.
• Injectivity of f alone is insufficient; a non‑injective g could still merge distinct f outputs.
• Surjectivity of either function does not affect the injectivity of the composition.

How to Remember

• Mnemonic: “Injective g Keeps h Unique.”
• Tip: Visualize the composition as two stages; only the second stage must preserve distinctness.

Solving Equality of Functions on Specific Domains

Finding the Domain Where Two Functions Coincide

Given f(x) = x² − 2x and g(x) = −x + 6, we ask: on which real domain does f equal g?

• Set the expressions equal: x² − 2x = −x + 6.
• Bring all terms to one side: x² − x − 6 = 0.
• Factor: (x − 3)(x + 2) = 0, giving solutions x = 3 and x = −2.
• Thus the domain where the two functions match is the set {‑2, 3}.

How to Remember

• Mnemonic: “Equal → Subtract → Zero → Factor.”
• Tip: For any “when do these equal?” problem, compute f − g = 0 and factor; the roots are the exact points of equality.

Relations That Define Functions

One Ordered Pair per Input Guarantees a Function

If a relation R ⊆ A × B contains exactly one ordered pair for each element of A, then R satisfies the definition of a function from A to B.

• Each input has a single output – the hallmark of a function.
• No additional requirements are imposed for injectivity or surjectivity.
• Therefore, the property is: R is a function from A to B.

How to Remember

• Mnemonic: “One‑to‑One Input, One‑to‑One Output.”
• Tip: Imagine a vending machine that gives exactly one snack for each button; that mapping is a function regardless of which snacks are repeated or omitted.

Finding Inverses of Linear Functions

Inverse of f(x) = 2x + 3

For a linear function f(x) = ax + b with a ≠ 0, the inverse swaps the roles of x and y and solves for the new y.

• Write y = 2x + 3.
• Isolate x: x = (y − 3) / 2.
• Thus the inverse function is f⁻¹(y) = (y − 3) / 2.

How to Remember

• Mnemonic: “Undo the shift, then undo the stretch.” First subtract the constant, then divide by the coefficient.
• Tip: Always start with y = f(x) and solve for x; the resulting expression is f⁻¹(y).

The Identity Function

Definition and Properties of I_A

The identity function on a set A, denoted I_A, is defined by I_A(x) = x for every x ∈ A.

• It maps each element to itself, so it is automatically injective (no two inputs share an output).
• Because every element of A appears as an output, it is also surjective.
• Consequently, I_A is a bijection.

How to Remember

• Mnemonic: “I = I’m myself.” The identity leaves everything unchanged.
• Tip: If a function leaves every element exactly as it is, it automatically satisfies both injectivity and surjectivity.

Summary of Core Concepts

• Relations are subsets of Cartesian products; the number of possible relations equals 2^{|A|·|B|}.
• Injectivity requires distinct inputs to have distinct outputs; a single counter‑example disproves it.
• Surjectivity demands that every element of the codomain be hit at least once.
• A strictly monotone real function is always injective, but its range determines surjectivity.
• For a composite g ∘ f to be injective, the outer function g must be injective.
• Equality of two functions on a domain reduces to solving f(x) – g(x) = 0.
• A relation with exactly one ordered pair per input is a function; additional properties depend on further constraints.
• The inverse of a linear function ax + b is (y – b)/a.
• The identity function I_A is the canonical bijection on a set.

Mastering these ideas equips you with the tools to tackle more advanced topics such as equivalence relations, cardinality, and functional analysis. Keep revisiting the mnemonics and tips—they are designed to turn abstract definitions into memorable patterns.