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Let us recall difference of two sets.
For any three sets \(A\), \(B\) and \(C\):
(i) \(A - (B \cup C)\) \(=\) \((A-B) \cap (A-C)\)
(ii) \(A - (B \cap C)\) \(=\) \((A-B) \cup (A-C)\)
Example:
1. Let \(A\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\), \(B\) \(=\) \(\{-1\), \(0\), \(1\), \(3\), \(4\}\) and \(C\) \(=\) \(\{-3\), \(-1\), \(1\), \(4\), \(5\}\).
Verify that \(A - (B \cup C)\) \(=\) \((A-B) \cap (A-C)\).
L.H.S: \(A - (B \cup C)\)
\(B \cup C\) \(=\) \(\{-1\), \(0\), \(1\), \(3\), \(4\}\) \(\cup\) \(\{-3\), \(-1\), \(1\), \(4\), \(5\}\)
\(B \cup C\) \(=\) \(\{-3\), \(-1\), \(0\), \(1\), \(3\), \(4\), \(5\}\)
\(A - (B \cup C)\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\) \(-\) \(\{-3\), \(-1\), \(0\), \(1\), \(3\), \(4\), \(5\}\)
\(A - (B \cup C)\) \(=\) \(\{-5\), \(-4\), \(2\}\) - - - - - (I)
R.H.S: \((A-B) \cap (A-C)\)
\(A-B\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\) \(-\) \(\{-1\), \(0\), \(1\), \(3\), \(4\}\)
\(A-B\) \(=\) \(\{-5\), \(-4\), \(2\}\)
\(A-C\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\) \(-\) \(\{-3\), \(-1\), \(1\), \(4\), \(5\}\)
\(A-C\) \(=\) \(\{-5\), \(-4\), \(0\), \(2\}\)
\((A-B) \cap (A-C)\) \(=\) \(\{-5\), \(-4\), \(2\}\) \(\cap\) \(\{-5\), \(-4\), \(0\), \(2\}\)
\((A-B) \cap (A-C)\) \(=\) \(\{-5\), \(-4\), \(2\}\) - - - - - (II)
From (I) and (II), we see that:
\(A - (B \cup C)\) \(=\) \((A-B) \cap (A-C)\).
Hence verified.
2. Let \(A\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\), \(B\) \(=\) \(\{-1\), \(0\), \(1\), \(3\), \(4\}\) and \(C\) \(=\) \(\{-3\), \(-1\), \(1\), \(4\), \(5\}\).
Verify that \(A - (B \cap C)\) \(=\) \((A-B) \cup (A-C)\).
L.H.S: \(A - (B \cap C)\)
\(B \cap C\) \(=\) \(\{-1\), \(0\), \(1\), \(3\), \(4\}\) \(\cap\) \(\{-3\), \(-1\), \(1\), \(4\), \(5\}\)
\(B \cap C\) \(=\) \(\{\)\(-1\), \(1\), \(4\)\(\}\)
\(A - (B \cap C)\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\) \(-\) \(\{\)\(-1\), \(1\), \(4\)\(\}\)
\(A - (B \cap C)\) \(=\) \(\{-5\), \(-4\), \(0\), \(2\}\) - - - - - (I)
R.H.S: \((A-B) \cup (A-C)\)
\(A-B\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\) \(-\) \(\{-1\), \(0\), \(1\), \(3\), \(4\}\)
\(A-B\) \(=\) \(\{-5\), \(-4\), \(2\}\)
\(A-C\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\) \(-\) \(\{-3\), \(-1\), \(1\), \(4\), \(5\}\)
\(A-C\) \(=\) \(\{-5\), \(-4\), \(0\), \(2\}\)
\((A-B) \cup (A-C)\) \(=\) \(\{-5\), \(-4\), \(2\}\) \(\cup\) \(\{-5\), \(-4\), \(0\), \(2\}\)
\((A-B) \cup (A-C)\) \(=\) \(\{-5\), \(-4\), \(0\), \(2\}\) - - - - - (II)
From (I) and (II), we see that:
\(A - (B \cap C)\) \(=\) \((A-B) \cup (A-C)\).
Hence verified.
Important!
L.H.S – Left Hand Side
R.H.S – Right Hand Side
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