Question

if A and B are two non empty sets and A×B=A×C..Then show that B=C

Answer 1

If A×B=A×C
B=C×A/A
B=C
Hence showed

Answer 2

Answer:

Let b be an arbitrary element of B. Then,

$\\ \sf \: ( \: a,b) \: \in \: A \times B\: \: for \: \: all \: \: a \: \in \: A \\ \\ \\ \implies\sf \: ( \: a,b) \: \in \: A \times C \: \: for \: \: all \: \: a \: \in \: A \\ \\ \\ \implies \sf \: b \: \in \: \: C$

Thus,

$\\ \sf \: b \: \in \: B \implies \: b \: \in \: C \\ \\ \\ \therefore \sf \qquad \blue{ B \: \subset \: C} \qquad \quad \qquad \: (1)$

Now, let c be an arbitrary element of C. Then,

$\\ \sf \: ( \: a,c) \: \in \: A \times C \: \: for \: \: all \: \: a \: \in \: \: A \\ \\ \\ \implies \sf \: ( \: a,c) \: \in \: A \times B \: \: for \: \: all \: \: a \: \in \: aA \\ \\ \\ \implies \sf \: C \: \in \: B$

Thus,

$\\ \sf \: C \: \in \: C \implies \: c \: \in \: B\\ \\ \\ \therefore \qquad \sf \green{C\: \subset \: B} \qquad \quad \qquad \: \: (2)\\ \\ \:$

From (1) and (2), we get B = C.