Kinds of relations

In Class 11 you met a relation from a set A to a set B: a set of ordered pairs (a, b). In Class 12 we mostly look at a relation in one set A, so R is a set of pairs taken from A × A.

Example: in A = {1, 2, 3}, let R be 'a + b > 0'. Every pair works, so R contains all nine pairs.

A set with n elements has n² ordered pairs in A × A, and a relation can include or leave out each one. So there are 2 to the power n² relations in A. With 3 elements that is 2⁹ = 512.

Two extreme kinds:

• Empty relation: no pair is related, so R = φ. In A = {1, 2, 3, 4}, 'a + b = 20' is empty: no two of these numbers add up to 20.
• Universal relation: every pair is related, so R = A × A. 'a + b > 0' above is universal.

Both are called trivial relations. Most useful relations sit in between.

Three properties decide what kind of relation R is:

• Reflexive: every element is related to itself: (a, a) is in R for every a in A.
• Symmetric: whenever (a, b) is in R, (b, a) is in R too. Think of birthday wishes between true friends: if a posts for b, b must post for a.
• Transitive: whenever (a, b) and (b, c) are in R, (a, c) is in R too. It is a chain rule: a to b, b to c, so a to c.

Do not confuse reflexive with the identity relation, which contains the pairs (a, a) for every a and nothing else. Every identity relation is reflexive, but a reflexive relation may contain other pairs as well.

Checking the properties in a list of pairs:

• Reflexive: look only for the 'doubles' (1, 1), (2, 2), ... One missing double and R is not reflexive.
• Symmetric: for every pair, find its reverse. Doubles are their own reverse, so they never cause trouble.
• Transitive: for every pair (a, b), look for pairs that start with b. If there are any, say (b, c), then (a, c) must be there. If no pair starts with b, there is nothing to check for (a, b): it passes.

Example: R = {(1, 2), (3, 4)} in {1, 2, 3, 4}. No pair starts with 2 and none starts with 4, so no chain ever forms: R is transitive. But it is not reflexive (no doubles) and not symmetric ((2, 1) is missing).

To show a property fails, one counter-example is enough. To show it holds, you must argue for every pair, usually with letters a, b, c rather than numbers.