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.
Figures 9.20(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect ? Why ?
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:
2. Intersection of Sets:
3.Set Difference:
4.Set Complement: