If A - B - C - D and A - B - prove that A - C and B - D-

Let (a, b) be an arbitary element of A × B then, (a, b) ϵ A × B ⇒ a ϵ A and b ϵ B ...(i) Now, (a, b) ϵ A × B ⇒ (a, b) ϵ C × D [∵ A × B ⊆ C × D] ⇒ a ϵ ...

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Byju's Answer

Standard XII

Mathematics

Subset

If A × B ⊆ C ...

Question

If $A \times B \subseteq C \times D$ and $A \times B \neq ϕ,$ prove that $A \subseteq C a n d B \subseteq D .$

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Solution

Let (a, b) be an arbitary element of $A \times B$ then,

$(a, b) ϵ A \times B$

$\Rightarrow a ϵ A$ and $b ϵ B$ ...(i)

Now,

$(a, b) ϵ A \times B$

$\Rightarrow (a, b) ϵ C \times D$ $[∵ A \times B \subseteq C \times D]$

$\Rightarrow a ϵ C$ and $b ϵ D$ ...(ii)

$∴ a ϵ A \Rightarrow a ϵ C$ [Using (i) and (ii)]

$\Rightarrow A \subseteq C$

and,

$\Rightarrow b ϵ B \Rightarrow b ϵ D$

$\Rightarrow B \subseteq D$

Hence, proved.

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If A×B⊆C×D and A×B≠ϕ, prove that A⊆C and B⊆D.

Let (a, b) be an arbitary element of A×B then, (a,b)ϵA×B ⇒aϵA and bϵB ...(i) Now, (a,b)ϵA×B ⇒(a,b)ϵC×D [∵A×B⊆C×D] ⇒aϵC and bϵD ...(ii) ∴aϵA⇒aϵC [Using (i) and (ii)] ⇒A⊆C and, ⇒bϵB⇒bϵD ⇒B⊆D Hence, proved.

If $A \times B \subseteq C \times D$ and $A \times B \neq ϕ,$ prove that $A \subseteq C a n d B \subseteq D .$