Relations and Functions · Study notes

MATRelations

Types of Relations

A relation $R$ on a set $A$ is any subset of $A \times A$ , and we classify it by checking three properties: reflexive, symmetric and transitive. The headline result examined is the equivalence relation (all three hold), which partitions $A$ into disjoint equivalence classes.

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ISC questions almost always ask you to prove a relation is an equivalence relation and then list its distinct classes, so a clean property-by-property argument plus correct class-counting earns full marks.

Reflexive

(a,a) \in R \quad \forall a \in A

R \subseteq A \times A

; every element must be related to itself.

Symmetric

(a,b) \in R \Rightarrow (b,a) \in R \quad \forall a,b \in A

Whenever

a

is related to

b

b

must be related to

a

Transitive

(a,b) \in R \ \text{and}\ (b,c) \in R \Rightarrow (a,c) \in R

Holds

\forall a,b,c \in A

; an equivalence relation is reflexive, symmetric and transitive together.

Equivalence class

[a] = \{\, x \in A : (x,a) \in R \,\}

[a]

is the class of

a

; distinct classes are disjoint and their union is all of

A

(a partition).

Congruence modulo

n

a \equiv b \ (\mathrm{mod}\ n) \iff n \mid (a-b)

\mathbb{Z}

this is an equivalence relation with exactly

n

classes

[0],[1],\dots,[n-1]

(the remainders).

To prove an equivalence relation, verify reflexive, symmetric AND transitive separately and explicitly; missing one means it is not an equivalence relation.
For the divisibility relation $3 \mid (a-b)$ on $\mathbb{Z}$ , there are exactly $3$ classes $[0],[1],[2]$ (numbers leaving remainder $0,1,2$ on division by $3$ ); for $5 \mid (a-b)$ there are $5$ classes.
For $R = \{(a,b): |a-b| \text{ is even}\}$ on $\{1,2,3,4,5\}$ the two classes are the odds $\{1,3,5\}$ and the evens $\{2,4\}$ , since $|a-b|$ is even exactly when $a$ and $b$ have the same parity.
Equivalence classes form a partition: any two classes are equal or disjoint, and they cover the whole set with no overlaps.
A 'sameness' relation (same locality, same blood group, etc.) is automatically an equivalence relation, because 'sameness' is always reflexive, symmetric and transitive; each class collects all elements sharing that attribute.
Parallelism of lines (taking each line as parallel to itself) is an equivalence relation; each class is a set of mutually parallel lines (one common direction/slope).
To show a relation is NOT an equivalence relation, give ONE explicit counterexample for the property that fails — e.g. for $R: a \le b^2$ on $\mathbb{R}$ , reflexivity fails at $a = \tfrac{1}{2}$ since $\tfrac{1}{2} \le \tfrac{1}{4}$ is false.

Where the marks go

Confusing symmetric with antisymmetric, or asserting transitivity from a single chain — you must show it holds for ALL applicable $a,b,c$ , not one example.
For $R: a \le b^2$ on $\mathbb{R}$ , students wrongly claim reflexivity; check $a=\tfrac{1}{2}$ , where $a \le a^2$ becomes $\tfrac{1}{2} \le \tfrac{1}{4}$ and fails, so $R$ is not even reflexive.
Miscounting equivalence classes: congruence modulo $n$ gives exactly $n$ classes (the distinct remainders), not infinitely many — list them as $[0],[1],\dots,[n-1]$ .
Writing equivalence classes that overlap; correct classes must be mutually disjoint and together exhaust the set $A$ .

How the board asks it

Derive / provereflexive, symmetric and transitive
Show that the relation $R$ on the set $\mathbb{Z}$ of integers, defined by $R = \{(a,b) : 3 \mid (a-b)\}$ , is an equivalence relation. Hence write down the equivalence class $[0]$ .
Numericalequivalence classes as a partition
The relation $R$ on the set $A = \{1,2,3,4,5\}$ is given by $R = \{(a,b) : |a-b| \text{ is even}\}$ . Find all the distinct equivalence classes of $R$ and state how many there are.
Give reasonsone explicit counterexample for the property that fails
Examine whether the relation $R$ on $\mathbb{R}$ defined by $R = \{(a,b) : a \le b^2\}$ is reflexive, symmetric or transitive. Give reasons for each property, supporting your answer with a counterexample where it fails.
Applicationa 'sameness' relation
Let $R$ be the relation on the set of all human beings in a town defined by ' $a$ is related to $b$ if $a$ and $b$ have the same blood group'. Show that $R$ is an equivalence relation, and describe its equivalence classes.

Practise this topic

NextTypes of Functions (One-One, Onto, Bijective)

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.