UPSKILL MATH PLUS
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Learn moreLet us recall the union and intersection of two sets.
Union of two sets:
The union of two sets \(A\) and \(B\) is the set of all elements which are either in \(A\) or in \(B\) or in both. It is denoted by \(A\cup B\) and read as '\(A\) union \(B\)'.
Intersection of two sets:
The intersection of two sets \(A\) and \(B\) is the set of all elements common to both \(A\) and \(B\). It is denoted by \(A\cap B\) and read as '\(A\) intersection \(B\)'.
Commutative property for union of two sets:
The union of two sets will not change if you interchange the order of the two sets.
\(A\cup B\) \(=\) \(B \cup A\)
Let \(A\) \(=\) \(\{\)\(2\), \(3\), \(5\), \(7\)\(\}\), and \(B\) \(=\) \(\{\)\(4\), \(6\), \(8\), \(10\)\(\}\)
\(A \cup B\) \(=\) \(\{\)\(2\), \(3\), \(5\), \(7\)\(\}\) \(\cup\) \(\{\)\(4\), \(6\), \(8\), \(10\)\(\}\)
\(A \cup B\) \(=\) \(\{\)\(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(10\)\(\}\)
\(B \cup A\) \(=\) \(\{\)\(4\), \(6\), \(8\), \(10\)\(\}\) \(\cup\) \(\{\)\(2\), \(3\), \(5\), \(7\)\(\}\)
\(B \cup A\) \(=\) \(\{\)\(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(10\)\(\}\)
From the above results, we see that:
\(A \cup B\) \(=\) \(B \cup A\)
This is known as commutative property of union of two sets.
Commutative property for intersection of two sets
The intersection of two sets will not change if you interchange the order of the two sets.
\(A\cap B\) \(=\) \(B \cap A\)
Let \(A\) \(=\) \(\{\)\(3\), \(5\), \(8\), \(9\), \(10\)\(\}\), and \(B\) \(=\) \(\{\)\(4\), \(5\), \(7\), \(10\)\(\}\)
\(A \cap B\) \(=\) \(\{\)\(3\), \(5\), \(8\), \(9\), \(10\)\(\}\) \(\cap\) \(\{\)\(4\), \(5\), \(7\), \(10\)\(\}\)
\(A \cap B\) \(=\) \(\{\)\(5\), \(10\)\(\}\)
\(B \cap A\) \(=\) \(\{\)\(4\), \(5\), \(7\), \(10\)\(\}\) \(\cap\) \(\{\)\(3\), \(5\), \(8\), \(9\), \(10\)\(\}\)
\(B \cap A\) \(=\) \(\{\)\(5\), \(10\)\(\}\)
From the above results, we see that:
\(A \cap B\) \(=\) \(B \cap A\)
This is known as commutative property of intersection of two sets.
Important!
The generalized meaning of the commutative property is the result will not change if you interchange the order of the sets.
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