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1.Set Theory
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मान लीजिए कि $A$ और $B$ समुचचय हैं। यदि किसी समुचचय $X$ के लिए $A \cap X = B \cap X =\phi$ तथा $A \cup X = B \cup X ,$ तो सिद्ध कीजिए कि $A = B$.
Option A
Option B
Option C
Option D
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 $A = B$
Standard 11
Mathematics