Question

Let U be the universal set and $A\cup B\cup C=U$. Then $(A-B)\cup (B-C)\cup (C-A)$ = ____

• A. $A\cup B\cup C$
• B. $A\cup (B\cap C)$
• C. $(A\cap B\cap C{)}^{{}^{\prime }}$
• D. $A\cap (B\cup C)$

Answer 1

The correct option is C

$(A\cap B\cap C{)}^{{}^{\prime }}$

$(A-B)\cup (B-C)\cup (C-A)$ = $(A\cap B\cap C{)}^{{}^{\prime }}$

Lets explain with an example

Let $U$ = $\{1,2,3,4,5,6,7,8,9,10\}$

$A$ = $\{2,4,8,5\}$

$B$ = $\{1,2,3,6,10\}$

$C$ = $\{2,3,7,8,9\}$

We can see that $A\cup B\cup C=U$

$A-B$ = $\{4,8,5\}$

$B-C$ = $\{1,6,10\}$

$C-A$ = $\{3,7,9\}$

Then, {$(A-B)\cup (B-C)\cup (C-A)$} = $\{1,3,4,5,6,7,8,9,10\}$ ...........(i)

$(A\cap B\cap C)$ = $\left\{2\right\}$

$\therefore$ $(A\cap B\cap C{)}^{{}^{\prime }}$ = $\{1,3,4,5,6,7,8,9,10\}$...........(ii)

From (i) and (ii)

$(A-B)\cup (B-C)\cup (C-A)$ = $(A\cap B\cap C{)}^{{}^{\prime }}$