Subjective Type

For any sets A,B and C, prove that: A×(BC)=(A×B)(A×C)

Solution

Let A={3,1}, B={1,3} and C={3,5}

1) BC={1,3}{3,5}
BC={1,3,5}

A×(BC)={3,1}×{1,3,5}

A×(BC)={(3,1),(3,3),(3,5),(1,1),(1,3),(1,5)} Equation (1)

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

A×C={3,1}×{3,5}
A×C={(3,3),(3,5),(1,3),(1,5)}

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

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

From equation (1) and (2),
A×(BC)=(A×B)(A×C)


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