1.Set TheoryMedium

$A$ અને $B$ ગણો છે. કોઈ ગણ $X$ માટે જો $A \cap X=B \cap X=\phi$ અને $A \cup X=B \cup X$ તો સાબિત કરો કે $A = B$

( સૂચનઃ $A = A \cap (A \cup X),B = B \cap (B \cup X)$ અને વિભાજનના નિયમનો ઉપયોગ કરો. )

Solution

Let $A$ and $B$ be two sets such that $A \cap X=B \cap x=f$ and $A \cup X=B \cup X$ for some

To show: $A=B$

It can be seen that

$A=A \cap(A \cup X)=A \cap(B \cup X)[A \cup X=B \cup X]$

$=(A \cap B) \cup(A \cap X)$ [Distributive law]

$=(A \cap B) \cup \varnothing[A \cap X=\varnothing]$

$=A \cap B$ .........$(1)$

Now, $B=B \cap(B \cup X)$

$=B \cap(A \cup X)[A \cup X=B \cup X]$

$=(B \cap A) \cup(B \cap X)$ [Distributive law]

$=(B \cap A) \cup \varnothing[B \cap X=\varnothing]$

$=B \cap A$

$=A \cap B$ ...........$(2)$

Hence, from $(1)$ and $(2),$ we obtain $A = B$

જો $A=\{3,6,9,12,15,18,21\}, B=\{4,8,12,16,20\},$ $C=\{2,4,6,8,10,12,14,16\}, D=\{5,10,15,20\} ;$ તો મેળવો : $B-A$

Easy

Easy

Difficult

Medium

Easy