Lesson 3 of 10 · 8 min
Relations
NCERT §2.3
The shop cannot supply every one of its 8 kinds. Only navy in M and L, and white in S, M and L, are in stock. The in-stock list is still a set of pairs, just a smaller one.
The lesson in notes
In short
Take non-empty sets A and B. Any subset R of A × B is a relation from A to B; usually a rule linking the first entry x to the second entry y decides which pairs belong. (x, y) ∈ R is also written x R y.
If (x, y) ∈ R, then y is called an image of x under R.
The domain of R is the set of first entries of its pairs, and the range is the set of second entries. B is the codomain, and range ⊂ codomain; the range can be strictly smaller.
A relation can be given in roster form, in set-builder form or by an arrow diagram. For A = {1, 2, 3, 4, 5} and B = {1, 2, …, 10}, R = {(x, y) : y = 2x} is {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}, with domain A, range {2, 4, 6, 8, 10} and codomain B.
In a relation an element of A may have several images, and it may have none; such an element is simply missing from the domain.
Since a relation is a subset of A × B, and A × B has pq elements when n(A) = p and n(B) = q, there are 2^(pq) relations from A to B, counting φ and A × B itself.
A relation from A to A is called a relation on A.