Class 11th Maths Textbook Chapter 1: Sets Exercise 6: Miscellaneous Question 9 Solution

Question

Let A and B be sets. If A $\cap$ X = B $\cap$ X = $\emptyset$ and A $\cup$ X = B $\cup$ X for some set X, show that A = B. (Hint: A = A $\cap$ (A $\cup$ X), B = B $\cap$ (B $\cup$ X) and use Distributive law.)

Accepted Answer

Start with the identity A = A $\cap$ (A $\cup$ X) and use the given condition A $\cup$ X = B $\cup$ X
 	ext{A = A $\cap$ (A $\cup$ X) = A $\cap$ (B $\cup$ X)} Apply the Distributive law and use the fact that A and X are disjoint.
 	ext{A = A $\cap$ (B $\cup$ X) = (A $\cap$ B) $\cup$ (A $\cap$ X)}|||	ext{Since A $\cap$ X = $\emptyset$, A = (A $\cap$ B) $\cup$ $\emptyset$ = A $\cap$ B} Start with the identity B = B $\cap$ (B $\cup$ X) and use the given condition A $\cup$ X = B $\cup$ X
 	ext{B = B $\cap$ (B $\cup$ X) = B $\cap$ (A $\cup$ X)} Apply the Distributive law and use the fact that B and X are disjoint
 	ext{B = B $\cap$ (A $\cup$ X) = (B $\cap$ A) $\cup$ (B $\cap$ X)}|||	ext{Since B $\cap$ X = $\emptyset$, B = (B $\cap$ A) $\cup$ $\emptyset$ = B $\cap$ A} Compare the expressions obtained for A and B.
 {green}	extbf{Answer: Since A = A $\cap$ B and B = B $\cap$ A = A $\cap$ B, both A and B are equal to A $\cap$ B. Therefore A = B.}{/green}

Question: 9

Answer