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1.Set Theory
medium
ગણ $A, B$ અને $C$ માટે $A \cup B=A \cup C$ અને $A \cap B=A \cap C$ છે. સાબિત કરો કે, $B = C$.
Option A
Option B
Option C
Option D
Solution
Let, $A, B$ and $C$ be the sets such that $A \cup B=A \cup C$ and $A \cap B=A \cap C$.
To show: $B = C$
Let $x \in B$
$\Rightarrow x \in A \cup B \quad[B \subset A \cup B]$
$\Rightarrow x \in A \cup C \quad[A \cup B=A \cup C]$
$\Rightarrow x \in A$ or $x \in C$
Case $I$
Also, $x \in B$
$\therefore x \in A \cap B$
$\Rightarrow x \in A \cap C \quad[\because A \cap B=A \cap C]$
$\therefore x \in A$ and $x \in C$
$\therefore x \in C$
$\therefore B \subset C$
Similarly, we can show that $C \subset B$
$\therefore B=C$
Standard 11
Mathematics
