Sets and Functions

The One Thing

Sets provide a precise framework for grouping objects and counting relationships, while functions are relations that assign exactly one output to each input. Trigonometric functions extend right-triangle ratios to all real angles through the unit circle, with values governed by identities, quadrant signs, domains, and periods.

Definitions and Results

• Set: A well-defined collection of distinct objects called elements. A set is usually denoted by a capital letter, with its elements written inside braces.
• Roster form: A method of describing a set by listing its elements inside braces, such as $\{1,2,3\}$.
• Set-builder form: A method of describing a set by a property satisfied by all its elements, such as $\{x:x\text{ is a natural number less than }4\}$.
• Empty set: A set containing no elements, represented by $\varnothing$ or $\{\}$.
• Finite and infinite sets: A finite set has a limited number of elements; an infinite set has endlessly many elements.
• Cardinality: The number of elements in a finite set $A$, written $n(A)$ or $|A|$.
• Subset: $A\subseteq B$ if every element of $A$ also belongs to $B$.
• Power set: $P(A)$ is the set of all subsets of $A$. If $A$ has $n$ elements, then $P(A)$ has $2^n$ elements and $A$ has $2^n-1$ proper subsets.
• Universal set: The set containing all objects under consideration in a particular discussion.
• Union: $A\cup B$ contains elements belonging to $A$, $B$, or both.
• Intersection: $A\cap B$ contains elements common to both $A$ and $B$.
• Difference of sets: $A-B$ consists of elements in $A$ that are not in $B$.
• Complement: $A^{\prime }$ or $A^c$ contains elements of the universal set that are not in $A$.
• Inclusion–exclusion for two finite sets:

$$n(A\cup B)=n(A)+n(B)-n(A\cap B).$$

• Inclusion–exclusion for three finite sets:

$$n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C).$$

• De Morgan’s laws:

$$(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime },(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }.$$

• Cartesian product: $A\times B$ is the set of all ordered pairs $(a,b)$, where $a\in A$ and $b\in B$. For finite sets,

$$n(A\times B)=n(A)n(B).$$ Generally, $A\times B\ne B\times A$.

• Ordered pair: A pair $(a,b)$ in which order matters; generally, $(a,b)\ne (b,a)$.
• Relation: A relation from $A$ to $B$ is any subset of $A\times B$. For finite sets, the number of possible relations is

$$2^{n(A)n(B)}.$$

• Domain: The set of first components of the ordered pairs in a relation or function.
• Codomain: The set into which the outputs of a function are assigned.
• Range: The set of actual output values produced by a relation or function. For $f:A\to B$, the range is a subset of $B$.
• Function: A relation in which every element of the domain has exactly one image in the codomain. A function from a set $A$ with $m$ elements to a set $B$ with $n$ elements can be formed in $n^m$ ways.
• Image and pre-image: If $f(a)=b$, then $b$ is the image of $a$, and $a$ is a pre-image of $b$.
• One-one function: Different elements of the domain have different images.
• Many-one function: Two or more domain elements may have the same image.
• Into function: The range is a proper subset of the codomain.
• Onto function: The range is equal to the codomain.
• Identity function: $f(x)=x$ for every element $x$ in its domain.
• Constant function: A function giving the same output for every input, such as $f(x)=c$.
• Composition of functions:

$$(f\circ g)(x)=f(g(x)),$$ where $g$ is applied first and then $f$. It is defined only when the output values of $g$ lie in the domain of $f$. Generally, $$f\circ g\ne g\circ f.$$

• Graphical test for a function: A graph represents a function only if every vertical line intersects it at most once.
• Function-domain restriction: For $f(x)=1/x$, $x=0$ is excluded. Restrictions may arise from zero denominators or even roots of negative numbers.
• Trigonometric ratios: For an acute angle in a right triangle,

$$\sin \theta =\frac{\text{perpendicular}}{\text{hypotenuse}},\cos \theta =\frac{\text{base}}{\text{hypotenuse}},\tan \theta =\frac{\text{perpendicular}}{\text{base}}.$$

• Reciprocal trigonometric functions:

$$\mathrm{cosec}\theta =\frac{1}{\sin \theta },\sec \theta =\frac{1}{\cos \theta },\cot \theta =\frac{1}{\tan \theta }.$$

• Radian measure: An angle measured as the ratio of arc length to radius.

$$180^{\circ }=\pi \text{ radians}.$$

• Unit circle: A circle of radius $1$ centred at the origin, used to define trigonometric functions for all real angles.
• Unit-circle coordinates: For the point corresponding to $\theta$, $\cos \theta$ is the $x$-coordinate and $\sin \theta$ is the $y$-coordinate.
• Trigonometric identities: Equations true for all values for which both sides are defined:

$${\sin }^2\theta +{\cos }^2\theta =1,$$ $$1+{\tan }^2\theta ={\sec }^2\theta ,$$ $$1+{\cot }^2\theta ={\mathrm{cosec}}^2\theta .$$

• Quotient identities: Wherever defined,

$$\tan \theta =\frac{\sin \theta }{\cos \theta },\cot \theta =\frac{\cos \theta }{\sin \theta }.$$

• Angle conversions:

$$180^{\circ }=\pi \text{ radians},360^{\circ }=2\pi \text{ radians},$$ $$1^{\circ }=\frac{\pi }{180}\text{ radians},1\text{ radian}=\frac{180^{\circ }}{\pi }.$$

• Quadrant signs: $\sin \theta$ is positive in quadrants I and II; $\cos \theta$ is positive in quadrants I and IV; $\tan \theta$ is positive in quadrants I and III.
• Periods: $\sin x$ and $\cos x$ have period $2\pi$; $\tan x$ has period $\pi$.
• Principal ranges:

$$-1\le \sin x\le 1,-1\le \cos x\le 1.$$ The function $\tan x$ can take all real values where it is defined.

• Undefined trigonometric functions: $\tan \theta$ and $\sec \theta$ are undefined when $\cos \theta =0$; $\cot \theta$ and $\mathrm{cosec}\theta$ are undefined when $\sin \theta =0$.

Worked Methods

Describing sets

• Identify the objects belonging to the collection.
• Use roster form when listing the elements explicitly, for example $\{1,2,3\}$.
• Use set-builder form when describing a common property, for example $\{x:x\text{ is a natural number less than }4\}$.
• If no objects satisfy the defining condition, write $\varnothing$ or $\{\}$.

Counting elements of unions

For two finite sets:

• Find $n(A)$.
• Find $n(B)$.
• Find $n(A\cap B)$.
• Substitute into

$$n(A\cup B)=n(A)+n(B)-n(A\cap B).$$

For three finite sets:

• Add $n(A)$, $n(B)$, and $n(C)$.
• Subtract $n(A\cap B)$, $n(B\cap C)$, and $n(C\cap A)$.
• Add $n(A\cap B\cap C)$.
• Use

$$n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C).$$

Constructing a Cartesian product

• Take each element $a$ of $A$.
• Pair it with every element $b$ of $B$.
• Write the resulting ordered pairs $(a,b)$.
• Remember that order matters, so $A\times B$ generally differs from $B\times A$.
• For finite sets, check the number of pairs using

$$n(A\times B)=n(A)n(B).$$

Classifying a relation or function

• Identify the ordered pairs.
• The first components form the domain.
• The assigned output set is the codomain.
• The actual second components form the range.
• Check whether every domain element has exactly one output:

- if yes, the relation is a function; - if not, it is not a function.

• Compare the range with the codomain:

- different inputs with different images give a one-one function; - repeated images give a many-one function; - a proper range gives an into function; - range equal to codomain gives an onto function.

For $f:A\to B$, the domain is $A$, the codomain is $B$, and the range is a subset of $B$.

Testing a graph

• Draw or inspect the graph.
• Consider vertical lines through the graph.
• If any vertical line intersects the graph more than once, the graph is not a function.
• If every vertical line intersects it at most once, it represents a function.

Finding a composite function

To calculate $(f\circ g)(x)$:

• Apply $g$ to $x$, obtaining $g(x)$.
• Substitute $g(x)$ into $f$.
• Simplify $f(g(x))$.
• Confirm that $g(x)$ lies in the domain of $f$.

The order is essential: $$(f\circ g)(x)=f(g(x)),$$ and generally $f\circ g\ne g\circ f$.

Applying right-triangle trigonometric ratios

For an acute angle $\theta$ in a right triangle:

• Identify the perpendicular, base, and hypotenuse relative to $\theta$.
• Use

$$\sin \theta =\frac{\text{perpendicular}}{\text{hypotenuse}},$$ $$\cos \theta =\frac{\text{base}}{\text{hypotenuse}},$$ $$\tan \theta =\frac{\text{perpendicular}}{\text{base}}.$$

• Use reciprocal functions when required:

$$\mathrm{cosec}\theta =\frac{1}{\sin \theta },\sec \theta =\frac{1}{\cos \theta },\cot \theta =\frac{1}{\tan \theta }.$$

Converting angle measures

• Use $180^{\circ }=\pi$ radians as the basic equivalence.
• For degrees to radians, multiply by $\pi /180$:

$$1^{\circ }=\frac{\pi }{180}\text{ radians}.$$

• For radians to degrees, multiply by $180/\pi$:

$$1\text{ radian}=\frac{180^{\circ }}{\pi }.$$

• Remember:

$$360^{\circ }=2\pi \text{ radians}.$$

Using the unit circle

• Locate the angle $\theta$ on the unit circle.
• Identify the corresponding point.
• Read the $x$-coordinate as $\cos \theta$.
• Read the $y$-coordinate as $\sin \theta$.
• Determine signs from the quadrant.
• Obtain $\tan \theta$ from

$$\tan \theta =\frac{\sin \theta }{\cos \theta },$$ provided $\cos \theta \ne 0$.

• Apply the appropriate period when interpreting repeated values:

$$\sin (x+2\pi )=\sin x,\cos (x+2\pi )=\cos x,\tan (x+\pi )=\tan x.$$

Simplifying trigonometric expressions

• Check that every expression is defined.
• Use

$$\tan \theta =\frac{\sin \theta }{\cos \theta },\cot \theta =\frac{\cos \theta }{\sin \theta }$$ where permitted.

• Apply

$${\sin }^2\theta +{\cos }^2\theta =1,$$ $$1+{\tan }^2\theta ={\sec }^2\theta ,$$ $$1+{\cot }^2\theta ={\mathrm{cosec}}^2\theta .$$

• Avoid cancelling or dividing by an expression that may be zero.
• State domain restrictions when denominators or even roots are involved.

Where It Goes Wrong

• Treating a relation as a function without checking that every domain element has exactly one output.
• Confusing codomain with range; the range may be a proper subset of the codomain, which determines whether a function is into or onto.
• Reversing the order in a Cartesian product or composition; generally $A\times B\ne B\times A$ and $f\circ g\ne g\circ f$.
• Forgetting that the composition $f\circ g$ is defined only when the output values of $g$ lie in the domain of $f$.
• Ignoring domain restrictions such as $x\ne 0$ for $f(x)=1/x$, or restrictions caused by zero denominators and even roots of negative numbers.
• Applying trigonometric identities without checking that both sides are defined, particularly where $\cos \theta =0$ or $\sin \theta =0$.

What Gets Asked

• Expressing sets in roster form, set-builder form, or using the empty set.
• Identifying finite and infinite sets, subsets, proper subsets, power sets, and cardinality.
• Calculating unions, intersections, differences, complements, and cardinalities using inclusion–exclusion.
• Applying De Morgan’s laws.
• Listing Cartesian products and determining their cardinalities.
• Identifying ordered pairs, relations, domains, codomains, and ranges.
• Determining whether a relation is a function, and classifying functions as one-one, many-one, into, onto, identity, or constant.
• Counting possible relations and functions for finite sets.
• Evaluating compositions and checking their domains.
• Testing whether a graph represents a function using the vertical-line test.
• Determining restrictions such as the excluded value $x=0$ in $f(x)=1/x$.
• Calculating right-triangle trigonometric ratios and reciprocal ratios.
• Converting between degrees and radians.
• Using the unit circle to determine sine and cosine coordinates, quadrant signs, periods, and ranges.
• Applying and rearranging

$${\sin }^2\theta +{\cos }^2\theta =1,1+{\tan }^2\theta ={\sec }^2\theta ,1+{\cot }^2\theta ={\mathrm{cosec}}^2\theta .$$

• Using

$$\tan \theta =\frac{\sin \theta }{\cos \theta },\cot \theta =\frac{\cos \theta }{\sin \theta }$$ while observing the required domain conditions.