PUMPA - THE SMART LEARNING APP

Take a 10 minutes test to understand your learning levels and get personalised training plan!

Download now on Google PlayAssociative 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 – L**eft Hand Side

**R.H.S – R**ight Hand Side