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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. A_{1}x A_{2}xA_{3} x . . . .x A_{n} = {≤ a_{1}, a_{2}, . . . , a_{n}): a_{1} ∈A_{1}, a_{2} ∈ A_{2}, . . . , a_{n} ∈ A_{n}}

*Illustration -:*

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*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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