If A - B- prove that A - A - A - B - B - A

Let (a, b) be an arbitrary element of A × A Then, (a,a)ϵ A × A ⇒ a ϵ A and a ϵ A ⇒ a ϵ A and a ϵ B [∵ A ⊆ B] ⇒ (a, b) ϵ A × B and (a, b) ϵ B × A ⇒ (a, b) ϵ ( ...

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Standard XI

Mathematics

Intersection

If A ⊆ B, pro...

Question

If $A \subseteq B$ , prove that $A \times A \subseteq (A \times B) \cap (B \times A)$

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Solution

Let (a, b) be an arbitrary element of $A \times A$

Then, $(a, a) ϵ A \times A \Rightarrow a ϵ A$ and $a ϵ A$

$\Rightarrow a ϵ A$ and $a ϵ B$ $[∵ A \subseteq B]$

$\Rightarrow (a, b) ϵ A \times B$ and $(a, b) ϵ B \times A$

$\Rightarrow (a, b) ϵ (A \times B) \cap (B \times A)$

$∴ A \times A \subseteq (A \times B) \cap (B \times A)$

Hence, $A \subseteq B \Rightarrow A \times A \subseteq (A \times B) \cap (B \times A)$ .

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If A⊆B, prove that A×A⊆(A×B)∩(B×A)

Let (a, b) be an arbitrary element of A×A Then, (a,a)ϵA×A⇒aϵA and aϵA ⇒aϵA and aϵB [∵A⊆B] ⇒(a,b)ϵA×B and (a,b)ϵB×A ⇒(a,b)ϵ(A×B)∩(B×A) ∴A×A⊆(A×B)∩(B×A) Hence, A⊆B⇒A×A⊆(A×B)∩(B×A).

If $A \subseteq B$ , prove that $A \times A \subseteq (A \times B) \cap (B \times A)$