## NCERT Solutions Class 11 Maths – Cartesian Product of Sets

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** Cartesian Product of Sets**

**Ordered Pair:**As the name indicates it represents a pair of elements written in small brackets and grouped together in a particular order, i.e., (a,b), a ∈A and b ∈ B where A and B are two sets.

**Cartesian Product of Sets: **Given two non-empty sets A and B, the set of all ordered pairs (a, b),where a ∈A and b ∈B is called Cartesian product of A and B. In symbols,we write

A × B = {(a, b) | a ∈A and b ∈B}

Example: If A = {1, 2, 3} and B = {5,6}, then

A × B = {(1, 5), (2, 5), (3, 5), (1, 6), (2, 6), (3, 6)}

**Some Results**

** **

- Two ordered pairs are equal, if and only if their corresponding first element is equal and the second element is also equal e. (x, y) = (a, b) if and only if x = a and y = b.

- If there are p elements in A and q elements in B, then there will be pqelements in A × B, i.e., if n(A) = p and n(B) = q, then n(A × B) = pq.

- If A and B are two non-empty sets and either A or B is an infinite set, then so is

A × B.

- A × A × A = {(a, b, c) : a, b, c ∈A}. Here (a, b, c) is called an ordered triplet.

- Cartesian Product is anti-commutative i.e A × B ≠ B × A.

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**Question**: Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine

(i) A × B (ii) B × A

(iii) Is A × B = B × A (iv) Is *n *(A × B) = *n *(B × A) ?

** Solution:**Since A = {1, 2, 3, 4} and B = {5, 7, 9}. Therefore,

(i) A × B = {(1, 5), (1, 7), (1, 9), (2, 5), (2, 7),

(2, 9), (3, 5), (3, 7), (3, 9), (4, 5), (4, 7), (4, 9)}

(ii) B × A = {(5, 1), (5, 2), (5, 3), (5, 4), (7, 1), (7, 2),

(7, 3), (7, 4), (9, 1), (9, 2), (9, 3), (9, 4)}

(iii) No, A × B ≠B × A. Since A × B and B × A do not have exactly the same

ordered pairs.

(iv) n (A × B) = n (A) × n (B) = 4 × 3 = 12

n (B × A) = n (B) × n (A) = 4 × 3 = 12

Hence n (A × B) = n (B × A)

**Question**: Find *x *and *y *if (*x *– *y*, *x *+ *y*) = (4, 10)

** Solution: ***x *– *y *= 4

* x *+ *y *= 10

On solving , 2*x *= 14

*x *= 7

and 7 – *y *= 4

y=3

**Question**: If A= {1, 2}, form the set A × A × A.

**Solution:**We have, A × A × A= {(1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1),

(2,2,2)}.

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