Question

Let A , B, C be sets such that A $\cap$ B $\ne \varphi$, B $\cap$ C $\ne \varphi$ and A $\cap$ C $\ne \varphi$ . Do you claim that A $\cap$ B $\cap$ C $\ne \varphi$? Justify your answer.

Answer 1

True, if $\mathrm{A}\cap \mathrm{B}\ne \mathrm{\varphi }$, $\mathrm{B}\cap \mathrm{C}\ne \mathrm{\varphi }$ and $\mathrm{A}\cap \mathrm{C}\ne \mathrm{\varphi }$, then $\mathrm{A}\cap \mathrm{B}\cap \mathrm{C}\ne \mathrm{\varphi }$

We can justify this with the help of a Venn diagram as follows:

$\mathrm{A}\cap \mathrm{B}\ne \mathrm{\varphi }$



$\mathrm{B}\cap \mathrm{C}\ne \mathrm{\varphi }$




$\mathrm{A}\cap \mathrm{C}\ne \mathrm{\varphi }$



i.e., $\mathrm{A}\cap \mathrm{B}\cap \mathrm{C}\ne \mathrm{\varphi }$, which is shown in following Venn diagram: