Relations and Functions

Maths · Class 11

Simulation · Maths · Class 11

A × B is not B × A

"A × B is not B × A" is a free, interactive Class 11 Maths simulation for JEE, built on the NCERT section "Working with Cartesian products" in Relations and Functions. A grid with A's elements on one axis and B's on the other; every cell is a pair. Switching to B × A transposes the grid so the pairs visibly change although the count pq stays the same.

From the lesson Working with Cartesian products in Relations and Functions. Change the values and watch what happens.

A × B is not B × AMaths · Class 11

What it shows

NCERT §2.2

A grid with A's elements on one axis and B's on the other; every cell is a pair. Switching to B × A transposes the grid so the pairs visibly change although the count pq stays the same.

Controls

4 to try

  • Steppern(A) 1–4
  • Steppern(B) 1–4
  • ToggleA × B / B × A
  • Tapa cell to read its pair

Try this

Change the sets so that A × B = B × A as sets (the sim confirms only when A = B), with n(A × B) = 9.

Common mistake

Writing A × B = B × A because both have pq elements.

The idea behind it

NCERT §2.2

  • Order matters: A × B and B × A contain different pairs in general, so A × B ≠ B × A, although n(A × B) = n(B × A). For P = {a, b} and Q = {5}, P × Q = {(a, 5), (b, 5)} while Q × P = {(5, a), (5, b)}.
  • For non-empty A and B, A × B = B × A holds only when A = B.
  • If A and B are non-empty and at least one of them is infinite, then A × B is infinite; for example N × {1} has the infinitely many pairs (1, 1), (2, 1), (3, 1), ….
  • The product distributes over intersection and union: A × (B ∩ C) = (A × B) ∩ (A × C) and A × (B ∪ C) = (A × B) ∪ (A × C). With A = {1, 2}, B = {2, 3} and C = {3, 4}, both sides of the first identity equal {(1, 3), (2, 3)}.
  • The factors can be read back from the product: A is the set of first entries of A × B and B the set of second entries. If A × A has 4 elements and contains (2, 5), then 2, 5 ∈ A and n(A) = 2, so A = {2, 5} and A × A = {(2, 2), (2, 5), (5, 2), (5, 5)}.
  • When n(A × B) is given together with a few pairs, use n(A × B) = n(A) × n(B) to fix how many elements each factor has before listing them.

Questions

3 answers

What does the "A × B is not B × A" simulation show?
A grid with A's elements on one axis and B's on the other; every cell is a pair. Switching to B × A transposes the grid so the pairs visibly change although the count pq stays the same.
What mistake does the "A × B is not B × A" simulation fix?
A common mistake it clears up: writing A × B = B × A because both have pq elements.
Is this simulation free?
Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Working with Cartesian products" in Relations and Functions (Class 11 Maths).

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