prove that


If $A \times B \subseteq C \times D$ and $A \times B \neq \phi$, prove that $A \subseteq C$ and $B \subseteq D .$



Given: $A \times B \subseteq C \times D$ and $A \times B \neq \phi$

Need to prove: $A \subseteq C$ and $B \subseteq D$

Let us consider, $(x, y)^{\in}(A \times B) \cdots(1)$

$\Rightarrow(x, y)^{\in}(C \times D)[\operatorname{as} A \times B \subseteq C \times D] \cdots(2)$

From (1) we can say that,

$x \in_{A}$ and $y \in_{B \cdots}$ (a)

From (2) we can say that,

$x^{\in} C$ and $y \in D \ldots$ (b)

Comparing (a) and (b) we can say that

$\Rightarrow x^{\in}$ A and $x^{\in} C$

$\Rightarrow A \subseteq C$


$\Rightarrow y \in_{B}$ and $y \in_{D}$

$\Rightarrow B \subseteq D$ [Proved]


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