Lesson 1 of 10 · 8 min

Ordered pairs and the Cartesian product

NCERT §2.2

The trip T-shirt form has two boxes: colour first, then size. Bilal writes "M, navy" in the wrong order, and the shop sends him nothing. Why does the order of two words matter so much?

The story this chapter follows: Class XI-B's museum trip

Class XI-B is going by bus to a science museum 60 km east of the school, along a straight highway. Put the school gate at x = 0, count kilometres east as positive, and imagine a stone at every whole kilometre. The T-shirt order, the seating plan, the travel time and the bus's position on the highway supply every relation and function in this chapter.

The lesson in notes

In short

An ordered pair (a, b) is two objects written in a fixed order: a is the first entry and b the second. (a, b) and (b, a) are different pairs unless a = b.

Two ordered pairs are equal exactly when their first entries are equal and their second entries are equal: (a, b) = (c, d) ⇔ a = c and b = d. For example, (2p − 1, q + 3) = (5, 7) gives p = 3 and q = 4.

For non-empty sets A and B, the Cartesian product A × B is the set of every ordered pair whose first entry comes from A and second entry from B: A × B = {(a, b) : a ∈ A, b ∈ B}.

If A or B is the empty set, there is no pair to form, so A × φ = φ × A = φ.

Counting: when A has p elements and B has q, A × B has exactly pq pairs, because every one of the p first entries combines with each of the q second entries.

Triples work the same way: A × A × A collects every (a, b, c) whose three entries all come from A, and each such (a, b, c) is an ordered triplet. The coordinate plane is R × R, all pairs (x, y) of reals, and three-dimensional space is R × R × R.