UPSKILL MATH PLUS

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Learn moreLet \(A = \{20\), \(25\), \(26\), \(30\), \(31\), \(33\), \(35\}\), \(B = \{25\), \(30\), \(32\), \(34\), \(35\), \(36\}\) and \(C = \{15\), \(26\), \(30\), \(32\), \(33\), \(35\}\).

**(i)**Calculate the total number of common elements in all the three sets.

**(ii)**Calculate the total number of common elements in \(A\) and \(B\).

**(iii)**Calculate the total number of common elements in \(B\) and \(C\).

**(iv)**Calculate the total number of common elements in \(A\) and \(C\).

**(v)**Calculate the total number of elements in all the three sets.

**Solution**:

\(A = \{20\), \(25\), \(26\), \(30\), \(31\), \(33\), \(35\}\)

Number of elements in set \(A = n(A) = 7\)

\(B = \{25\), \(30\), \(32\), \(34\), \(35\), \(36\}\)

Number of elements in set \(B = n(B) = 6\)

\(C = \{15\), \(26\), \(30\), \(32\), \(33\), \(35\}\)

Number of elements in set \(C = n(C) = 6\)

**(i) Total number of common elements in all the three sets**:

\(A \cap B \cap C\) \(=\) \(\{30\), \(35\}\)

\(n(A \cap B \cap C) = 2\)

**(ii)**

**Total number**

**of common elements in**\(A\)

**and**\(B\):

\(A \cap B\) \(=\) \(\{25\), \(30\), \(35\}\)

\(n(A \cap B) = 3\)

**(iii)**

**Total number**

**of common elements in**\(B\)

**and**\(C\):

\(B \cap C\) \(=\) \(\{30\), \(32\), \(35\}\)

\(n(B \cap C) = 3\)

**(iv)**

**Total number**

**of common elements in**\(A\)

**and**\(C\):

\(A \cap C\) \(=\) \(\{26\), \(30\), \(33\), \(35\}\)

\(n(A \cap C) = 4\)

**(v) Total number of elements in all the three sets**:

\(n(A \cup B \cup C)\) \(= n(A) + n(B) + n(C)\) \(- n(A \cap B)\) \(- n(B \cap C)\) \(- n(A \cap C)\) \(+ n(A \cap B \cap C)\)

\(n(A \cup B \cup C)\) \(=\) \(7 + 6 + 6 - 3 - 3 - 4 + 2\)

\(n(A \cup B \cup C)\) \(= 11\)

Important!

If \(A\) and \(B\) are two finite sets, then:

**1**. \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\)

**2**. \(n(A - B) = n(A) - n(A \cap B)\)

**3**. \(n(B - A) = n(B) - n(A \cap B)\)

**4**. \(n(A^{\prime}) = n(U) - n(A)\)

**5**. \(n(U) = n(A) + n(A^{\prime})\)