CBSE Class 11 Maths Notes: Relations and Functions – Cartesian Product of Sets

The Cartesian product ≤ also known as the cross product) of two sets A and B, denoted by AxB ≤ in the same order) is the set of all ordered pairs ≤ x, y) such that x∈A and y∈B. What we mean by ordered pair is that the pair≤ a, b) is not the same the pair as ≤ b, a) unless a = b. It implies that AxB ≠ BxA in general. Also if A contains m elements and B contains n elements then AxB contains mxn elements.

Similarly we can define AxA = {≤ x, y); x∈A and y∈A}. We can also define cartesian product of more than two sets.

e.g.  A₁x A₂xA₃ x . . . .x A_n = {≤ a₁, a₂, . . . , a_n): a₁ ∈A₁, a₂ ∈ A₂, . . . , a_n ∈ A_n}

Illustration -:

If A = {a, b, c} and B = {b, c, d} then evaluate
i). A∪B, A∩B, A–B and B–A
ii). AxB and BxA

Solution:

i) A∪B = {x: x∈A or x∈B}= {a, b, c, d}
A∩B = {x: x∈A and x∈B}= {b, c}
A-B = {x: x∈A and x ∉ B}= {a}
B-A = {x: x∈B and x∉B}= {d}

ii) AxB = {≤ x, y): x∈A and y∈B}
= {≤ a, b), ≤ a, c), ≤ a, d), ≤ b, b), ≤ b, c), ≤ b, d), ≤ c, b), ≤ c, c), ≤ c, d)}
BxA = {≤ x, y):  x∈B and y∈A}
= {≤ b, a), ≤ b, b), ≤ b, c),≤ c, a), ≤ c, b), ≤ c, c),≤ d, a), ≤ d, b), ≤ d, c)}

Note that AxB ≠ BxA.

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