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Associative property for union of three sets
You can group three sets in any order, and the union of three sets will be the same.
 
\(A \cup (B \cup C)\) \(=\) \((A \cup B) \cup C\)
Let \(A\) \(=\) \(\{\)\(10\), \(11\), \(12\)\(\}\), \(B\) \(=\) \(\{11\), \(12\), \(13\)\(\}\) and \(C\) \(=\) \(\{\)\(12\), \(13\), \(14\)\(\}\).
 
L.H.S: \(A \cup (B \cup C)\)
 
\(B \cup C\) \(=\) \(\{11\), \(12\), \(13\)\(\}\) \(\cup\) \(\{\)\(12\), \(13\), \(14\)\(\}\)
 
\(B \cup C\) \(=\) \(\{\)\(11\), \(12\), \(13\), \(14\)\(\}\)
 
\(A \cup (B \cup C)\) \(=\) \(\{\)\(10\), \(11\), \(12\)\(\}\) \(\cup\) \(\{\)\(11\), \(12\), \(13\), \(14\)\(\}\)
 
\(A \cup (B \cup C)\) \(=\) \(\{\)\(10\), \(11\), \(12\), \(13\), \(14\)\(\}\) - - - - - - (I)
 
R.H.S: \((A \cup B) \cup C\)
 
\(A \cup B\) \(=\) \(\{\)\(10\), \(11\), \(12\)\(\}\) \(\cup\) \(\{11\), \(12\), \(13\)\(\}\)
 
\(A \cup B\) \(=\) \(\{\)\(10\), \(11\), \(12\), \(13\)\(\}\)
 
\((A \cup B) \cup C\) \(=\) \(\{\)\(10\), \(11\), \(12\), \(13\)\(\}\) \(\cup\) \(\{\)\(12\), \(13\), \(14\)\(\}\)
 
\((A \cup B) \cup C\) \(=\) \(\{\)\(10\), \(11\), \(12\), \(13\), \(14\)\(\}\)  - - - - - - (II)
 
From (I) and (II), we see that:
 
\(A \cup (B \cup C)\) \(=\) \((A \cup B) \cup C\)
 
This is called associative property of union of three sets.
Associative property for intersection of three sets
You can group three sets in any order, and the intersection of three sets will be the same.
 
\(A \cap (B \cap C)\) \(=\) \((A \cap B) \cap C\)
Let \(A\) \(=\) \(\{\)\(a\), \(b\), \(c\), \(d\)\(\}\), \(B\) \(=\) \(\{c\), \(d\), \(e\)\(\}\) and \(C\) \(=\) \(\{\)\(d\), \(e\), \(f\)\(\}\)
 
L.H.S: \(A \cap (B \cap C)\)
 
\(B \cap C\) \(=\) \(\{c\), \(d\), \(e\)\(\}\) \(\cap\) \(\{\)\(d\), \(e\), \(f\)\(\}\)
 
\(B \cap C\) \(=\) \(\{\)\(d\), \(e\)\(\}\)
 
\(A \cap (B \cap C)\) \(=\) \(\{\)\(a\), \(b\), \(c\), \(d\)\(\}\) \(\cap\)  \(\{\)\(d\), \(e\)\(\}\)
 
\(A \cap (B \cap C)\) \(=\) \(\{\)\(d\)\(\}\)  - - - - - - (I)
 
R.H.S: \((A \cap B) \cap C\)
 
\(A \cap B\) \(=\) \(\{\)\(a\), \(b\), \(c\), \(d\)\(\}\) \(\cap\) \(\{c\), \(d\), \(e\)\(\}\)
 
\(A \cap B\) \(=\) \(\{c\), \(d\)\(\}\)
 
\((A \cap B) \cap C\) \(=\) \(\{c\), \(d\)\(\}\) \(\cap\) \(\{\)\(d\), \(e\), \(f\)\(\}\)
 
\((A \cap B) \cap C\) \(=\) \(\{\)\(d\)\(\}\)  - - - - - - (II)
 
From (I) and (II), we see that:
 
\(A \cap (B \cap C)\) \(=\) \((A \cap B) \cap C\)
 
This is called associative property of intersection of three sets.
 
Important!
L.H.S – Left Hand Side
 
R.H.S – Right Hand Side