Why 'Is Taller Than' Fails a Test That 'Is Equal To' Passes

'Is equal to', 'is parallel to', 'is a sibling of' and 'is taller than' all link members of the same set, but they behave very differently. Checking three properties — reflexive, symmetric and transitive — reveals which relations split a set neatly into groups of alike elements.

This lesson covers relations on a set and special relations, the three properties, and equivalence relations.

**A relation on a set A is any subset of $A\times A$; the empty relation contains no pairs, the universal relation is the whole of $A\times A$, and the identity relation contains exactly the pairs $(a,a)$ for every a in A.

Special relations on $A=\{1,2,3\}$:

- Empty** — $R=\varphi$, for example the relation $a-b=10$
- Universal — $R=A\times A$, all 9 pairs, for example $|a-b|\le 2$
- Identity — $I_A=\{(1,1),(2,2),(3,3)\}$

Counting. A set with n elements has $2^{n^2}$ relations on it; for $n=3$ that is $2^9=512$.

Worked example. On $A=\{1,2,3,4\}$, let R relate a to b when $b=a+1$. Then $R=\{(1,2),(2,3),(3,4)\}$, with domain $\{1,2,3\}$ and range $\{2,3,4\}$.

An everyday example. 'Lives in the same city as', applied to students of a Delhi school who all live in Delhi, is the universal relation, since every pair qualifies.

The substance. The identity relation is not the universal relation — on a set with more than one element, the identity relation leaves out every pair of different elements.

**A relation R on A is reflexive if $(a,a)\in R$ for every a in A, symmetric if $(a,b)\in R$ always implies $(b,a)\in R$, and transitive if $(a,b)\in R$ and $(b,c)\in R$ always imply $(a,c)\in R$.

Testing each property:

- Reflexive — check that every element is related to itself
- Symmetric — for each pair, look for its reverse
- Transitive** — for each chain $(a,b)$, $(b,c)$, look for $(a,c)$
- One counter-example is enough to disprove a property

Worked example. On $A=\{1,2,3\}$, let $R=\{(1,1),(2,2),(3,3),(1,2),(2,1),(2,3)\}$.

- Reflexive — yes, all three pairs $(a,a)$ are present
- Symmetric — no, $(2,3)$ is present but $(3,2)$ is not
- Transitive — no, $(1,2)$ and $(2,3)$ are present but $(1,3)$ is not

Worked example 2. On the real numbers, 'a is less than or equal to b' is reflexive and transitive but not symmetric, since $2\le 5$ is true while $5\le 2$ is false.

An everyday example. 'Is taller than' among students in a class is transitive — if Asha is taller than Ravi and Ravi is taller than Meena, then Asha is taller than Meena — but it is neither reflexive nor symmetric.

The substance. A property can hold because there is nothing to check — the empty relation is symmetric and transitive, though it is not reflexive on a non-empty set.

A relation is an equivalence relation when it is reflexive, symmetric and transitive at the same time, and every equivalence relation divides its set into disjoint equivalence classes of mutually related elements.

Worked example. On the integers, let a be related to b when $a-b$ is divisible by 3.

- Reflexive — $a-a=0$, which is divisible by 3
- Symmetric — if $a-b=3k$, then $b-a=3(-k)$
- Transitive — if $a-b=3k$ and $b-c=3m$, then $a-c=3(k+m)$

So the relation is an equivalence relation. Its classes are $\{\dots ,-3,0,3,6,\dots \}$, $\{\dots ,-2,1,4,7,\dots \}$ and $\{\dots ,-1,2,5,8,\dots \}$ — the integers leaving remainders 0, 1 and 2 on division by 3.

Worked example 2. On lines in a plane, 'is parallel to' (counting each line as parallel to itself) is an equivalence relation, while 'is perpendicular to' is not — it is symmetric, but no line is perpendicular to itself.

Classes never overlap. Two equivalence classes are either identical or share no element, so together they partition the set.

An everyday example. 'Was born in the same month as', on the students of a school, is an equivalence relation that splits the school into birth-month groups.

The substance. All three properties must be proved in general — checking a few examples proves nothing, but a single failure is enough to rule a relation out.