Question

For any sets A, B, C using properties of sets, prove that: $A-\left( B\cap C \right)=\left( A-B \right)\cup \left( A-C \right)$.

Answer 1

Hint: We have to know the different formulas related to sets and we have to know the formula for difference of sets that is $A-B=A\cap {B}'$and we have to know the formula $A\cap \left( B\cup C \right)=(A\cap B)\cup (A\cap C)$. ‘$\cup$’ represents union of two or more sets.’ $\cap$’ represents the intersection of two sets.

Complete step-by-step answer:
We know that $A-B=A\cap {B}'$. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)
We also know that ${{\left( A\cap B \right)}^{\prime }}={A}'\cup {B}'$and $A\cap \left( B\cup C \right)=(A\cap B)\cup (A\cap C)$
$A-\left( B\cap C \right)=A\cap {{\left( B\cap C \right)}^{\prime }}$. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
$=A\cap \left( {B}'\cup {C}' \right)$. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)
= $\left( A\cap {B}' \right)\cup \left( A\cap {C}' \right)$. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)
$=\left( A-B \right)\cup \left( A-C \right)$
Hence proved.

Note: We can find the relation between two sets using the venn diagram. We can derive the relation between two sets used in this problem like the difference of two sets. A venn diagram is a diagram that shows all possible logical relations between finite collection of different sets.