1
Question
Let A and B be sets. If A ∩ X = B ∩ X = Φ and A ∪ X = B ∪ X for some set X, show that A = B. (Hints A = A ∩ (A ∪ X), B = B ∩ (B ∪ X) and use distributive law)
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Solution
It is given that,
A ∩ X = B ∩ X = ϕ (1)
And
A ∪ X = B ∪ X (2)
Let’s take,
A = A ∩ ( A ∪ X )
From equation (2),
A = A ∩ ( B ∪ X )
From distributive law,
A ∩ ( B ∪ X ) = ( A ∩ B ) ∪ ( A ∩ X )
From equation (1), the above equation becomes,
A= ( A ∩ B ) ∪ ϕ = ( A ∩ B ) (3)
Now, let’s take
B = B ∩ ( B ∪ X )
From equation (2),
B = B ∩ ( A ∪ X )
From distributive law,
B ∩ ( A ∪ X ) = ( B ∩ A ) ∪ ( B ∩ X )
From equation (1),
B = ( B ∩ A ) ∪ ϕ = ( B ∩ A ) = ( A ∩ B ) (4)
From equation (3) and (4),
A = B
Thus, A = B .
It is given that,
And
Let’s take,
From equation (2),
From distributive law,
From equation (1), the above equation becomes,
Now, let’s take
From equation (2),
From distributive law,
From equation (1),
From equation (3) and (4),
Thus,
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