Lesson 1 of 10 · 9 min

Empty and universal relations

NCERT §1.2

The relay squad for sports day wears jerseys 1 to 5. The coach asks for every pair of runners whose jersey numbers add up to 20. Nobody steps forward. Is that still an answer?

The story this chapter follows: Class XII-A's sports day

Class XII-A runs the school sports day: 40 students in jerseys numbered 1 to 40, four houses in the stands, eight lanes on the track and a points table on the scoreboard. Every relation and function in this chapter is one of the lists the organisers keep: who shares a house, who wears which number, who runs in which lane, and how many points each finish earns.

The lesson in notes

In short

A relation R from A to B is any subset of A × B, and (a, b) ∈ R is written a R b. The definition does not ask for a meaningful link between a and b; any chosen set of pairs is a relation.

A relation in (or on) a set A is a subset of A × A. When A has n elements, A × A has n² pairs, so there are 2^(n²) relations on A.

The empty relation on A is R = φ ⊂ A × A: it contains no pair at all, so nothing in A is related to anything.

The universal relation on A is R = A × A: every element of A is related to every element, itself included.

The empty and the universal relation are the two extreme relations on A, and both are called trivial relations.

Examples on A = {1, 2, 3, 4, 5}: R = {(a, b) : a + b = 20} is empty, because the largest possible sum is 5 + 5 = 10; R′ = {(a, b) : |a − b| ≤ 4} is universal, because no two elements differ by more than 4.

A relation is often stated by its rule alone, such as a R b ⇔ b − a = 1. On {1, 2, 3, 4} this rule picks out exactly (1, 2), (2, 3) and (3, 4).