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நினைவு கூர்க : கணங்களின் வித்தியாசம்
\(A\), \(B\) மற்றும் \(C\) ஏதேனும் மூன்று கணங்கள் என்க :
(i) \(A - (B \cup C)\) \(=\) \((A-B) \cap (A-C)\)
(ii) \(A - (B \cap C)\) \(=\) \((A-B) \cup (A-C)\)
Example:
1. \(A\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\), \(B\) \(=\) \(\{-1\), \(0\), \(1\), \(3\), \(4\}\) மற்றும் \(C\) \(=\) \(\{-3\), \(-1\), \(1\), \(4\), \(5\}\) எனில் \(A - (B \cup C)\) \(=\) \((A-B) \cap (A-C)\) என நிரூபிக்க .
\(A - (B \cup C)\)
\(B \cup C\) \(=\) \(\{-1\), \(0\), \(1\), \(3\), \(4\}\) \(\cup\) \(\{-3\), \(-1\), \(1\), \(4\), \(5\}\)
\(B \cup C\) \(=\) \(\{-3\), \(-1\), \(0\), \(1\), \(3\), \(4\), \(5\}\)
\(A - (B \cup C)\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\) \(-\) \(\{-3\), \(-1\), \(0\), \(1\), \(3\), \(4\), \(5\}\)
\(A - (B \cup C)\) \(=\) \(\{-5\), \(-4\), \(2\}\) - - - - - (I)
\((A-B) \cap (A-C)\)
\(A-B\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\) \(-\) \(\{-1\), \(0\), \(1\), \(3\), \(4\}\)
\(A-B\) \(=\) \(\{-5\), \(-4\), \(2\}\)
\(A-C\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\) \(-\) \(\{-3\), \(-1\), \(1\), \(4\), \(5\}\)
\(A-C\) \(=\) \(\{-5\), \(-4\), \(0\), \(2\}\)
\((A-B) \cap (A-C)\) \(=\) \(\{-5\), \(-4\), \(2\}\) \(\cap\) \(\{-5\), \(-4\), \(0\), \(2\}\)
\((A-B) \cap (A-C)\) \(=\) \(\{-5\), \(-4\), \(2\}\) - - - - - (II)
(I) மற்றும் (II),இல் இருந்து :
\(A - (B \cup C)\) \(=\) \((A-B) \cap (A-C)\).
நிரூபிக்கப்பட்டது .
2. \(A\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\), \(B\) \(=\) \(\{-1\), \(0\), \(1\), \(3\), \(4\}\) மற்றும் \(C\) \(=\) \(\{-3\), \(-1\), \(1\), \(4\), \(5\}\) எனில்\(A - (B \cap C)\) \(=\) \((A-B) \cup (A-C)\) என நிரூபிக்க.
\(A - (B \cap C)\)
\(B \cap C\) \(=\) \(\{-1\), \(0\), \(1\), \(3\), \(4\}\) \(\cap\) \(\{-3\), \(-1\), \(1\), \(4\), \(5\}\)
\(B \cap C\) \(=\) \(\{\)\(-1\), \(1\), \(4\)\(\}\)
\(A - (B \cap C)\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\) \(-\) \(\{\)\(-1\), \(1\), \(4\)\(\}\)
\(A - (B \cap C)\) \(=\) \(\{-5\), \(-4\), \(0\), \(2\}\) - - - - - (I)
\((A-B) \cup (A-C)\)
\(A-B\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\) \(-\) \(\{-1\), \(0\), \(1\), \(3\), \(4\}\)
\(A-B\) \(=\) \(\{-5\), \(-4\), \(2\}\)
\(A-C\) \(=\) \(\{-5\), \(-4\), \(-1\), \(0\), \(1\), \(2\}\) \(-\) \(\{-3\), \(-1\), \(1\), \(4\), \(5\}\)
\(A-C\) \(=\) \(\{-5\), \(-4\), \(0\), \(2\}\)
\((A-B) \cup (A-C)\) \(=\) \(\{-5\), \(-4\), \(2\}\) \(\cup\) \(\{-5\), \(-4\), \(0\), \(2\}\)
\((A-B) \cup (A-C)\) \(=\) \(\{-5\), \(-4\), \(0\), \(2\}\) - - - - - (II)
(I) மற்றும் (II), இல் இருந்து :
\(A - (B \cap C)\) \(=\) \((A-B) \cup (A-C)\).
நிரூபிக்கப்பட்டது .
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