Question

If A={1, 2. 3, 4, 6, 8. 10, 12). B= (2, 3,4, 6, 8, 10, 11), and C= (2.5, 9, 10. 12. 14), find the sets of elementsthat belong to both A AND B and A ANDC Then, find the set of elements that are common to A B, and C

Answer 1

The sets A, B and C are given as lists of elements:

`\begin{gathered} A=\mleft\lbrace1,2,3,4,6,8,10,12\mright\rbrace \\ B=\mleft\lbrace2,3,4,6,8,10,11\mright\rbrace \\ C=\mleft\lbrace2,5,9,10,12,14\mright\rbrace \end{gathered}`

The elements that belong to both A and B are said to belong to the intersection between both sets. It can be represented using a Venn Diagram:

The set of elements that belong to both A and B, is:

`A\cap B=\mleft\lbrace2,3,4,6,8,10\mright\rbrace`

Similarly, the set of elements that belog to both A and C, is:

`A\cap C=\mleft\lbrace2,10,12\mright\rbrace`

To find the set of elements that are common to A, B, and C, check for those elements that belong to all the sets:

`A\cap B\cap C=\mleft\lbrace2,10\mright\rbrace`

Therefore:

The set of elements that belong to both A and B, is:

`A\cap B=\mleft\lbrace2,3,4,6,8,10\mright\rbrace`

The set of elements that belong to both A and C, is:

`A\cap C=\mleft\lbrace2,10,12\mright\rbrace`

The set of elements that are common to A, B and C, is:

`A\cap B\cap C=\mleft\lbrace2,10\mright\rbrace`