Question

If A is a subset of B and C is a subset of D , then prove that A x C is a subset of B x D​

Answer 1

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Given

• A ⊂ B ⟺ {x ∈ A ⟹ x ∈ B}

Similarly,

• C ⊂ D ⟺ {x ∈ C ⟹ x ∈ D}

To Prove

• A × C ⊂ B × D.

That is, we want to prove the implication

( y , z ) ∈ A × C ⟹ ( y , z ) ∈ B × D.

Suppose that

( y , z) ∈ A × C.

This implies that

{ y ∈ A } ∧ { z ∈ C}.

Consequently,

{ y ∈ B } ∧ { z ∈ C }

is also True, since A ⊂ C and B ⊂ D.

Finally,

( y , z ) ∈ B × D.

Since y and z were arbitrary, we have proved that

A × C ⊂ B × D .

✌✌Proved

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Answer 2

A x C is a subset of B x D​

Given: A⊂B and C⊂D

To prove: A x C ⊂ B x D​

Let (a,c) ∈ AXC

⇒ a∈A and c∈C

⇒ a∈B and c∈D

Since A is the subset of B, so the elements of A is present in B, same goes with c∈D as set C is the subset of D

⇒(a,c) ∈ BxD

⇒ if (a,c)∈ AXC, then (a,c) ∈ BXD.

⇒AXC ⊂BXD

Hence proved

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