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Mathematics

Assertion (A) : Let A = {1, 2, 3, 4, 5, 6} and B = {1, 3, 5, 7, 9} then A ∩ B ⊆ A and A ∩ B ⊆ B, always true for every pair of two sets.

Reason (R) : For any sets A and B, we have A ∩ B ⊆ A and A ∩ B ⊆ B.

  1. Both A and R are correct, and R is the correct explanation for A.

  2. Both A and R are correct, and R is not the correct explanation for A.

  3. A is true, but R is false.

  4. A is false, but R is true.

Sets

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Answer

Given : A = {1, 2, 3, 4, 5, 6} and B = {1, 3, 5, 7, 9}

A ∩ B means this set contains elements common to both A and B.

A ∩ B = {1, 3, 5}

A ∩ B ⊆ A means {1, 3, 5} is subset of {1, 2, 3, 4, 5, 6}, which is true.

A ∩ B ⊆ B means {1, 3, 5} is subset of {1, 3, 5, 7, 9}, which is also true.

So, assertion (A) is true.

For any sets A and B, we have A ∩ B ⊆ A and A ∩ B ⊆ B.

This is a fundamental property of set theory :

A ∩ B = {x | x ∈ A and x ∈ B}

So any element of A ∩ B is automatically in both A and B, which implies A ∩ B ⊆ A and A ∩ B ⊆ B are true.

So, reason (R) is true and it clearly explains assertion.

Hence, option 1 is the correct option.

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