Question:

Let A and B be sets. If \(A ∩ X = B ∩ X = \phi\) 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)

Updated On: Oct 23, 2023
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Solution and Explanation

Let A and B be two sets such that \(A ∩ X = B ∩ X = f\) and \(A ∪ X = B ∪ X\) for some set X. 
To show: \(A = B \)
It can be seen that 

\(A = A ∩ (A ∪ X) = A ∩ (B ∪ X) [A ∪ X = B ∪ X] \)
\(= (A ∩ B) ∪ (A ∩ X) \)       \(   [\)Distributive law] = \((A ∩ B) ∪ \phi [A ∩ X = \phi] \)
\(= A ∩ B\) …………………………………………………………….. (1) 

Now, \(B = B ∩ (B ∪ X) \)
\(= B ∩ (A ∪ X) [A ∪ X = B ∪ X] \)
\(= (B ∩ A) ∪ (B ∩ X)\) [Distributive law] 
\(= (B ∩ A) ∪ \phi [B ∩ X = \phi]\)
\(= B ∩ A \)
\(= A ∩ B\) …………………………………………………………… (2) 

Hence, from (1) and (2), we obtain A = B. 

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Concepts Used:

Operations on Sets

Some important operations on sets include union, intersection, difference, and the complement of a set, a brief explanation of operations on sets is as follows:

1. Union of Sets:

  • The union of sets lists the elements in set A and set B or the elements in both set A and set B.
  • For example, {3,4} ∪ {1, 4} = {1, 3, 4}
  • It is denoted as “A U B”

2. Intersection of Sets:

  • Intersection of sets lists the common elements in set A and B.
  • For example, {3,4} ∪ {1, 4} = {4}
  • It is denoted as “A ∩ B”

3.Set Difference:

  • Set difference is the list of elements in set A which is not present in set B
  • For example, {3,4} - {1, 4} = {3}
  • It is denoted as “A - B”

4.Set Complement:

  • The set complement is the list of all elements present in the Universal set except the elements present in set A
  • It is denoted as “U-A”