Let $A, B,$ and $C$ be the sets such that $A \cup B=A \cup C$ and $A \cap B=A \cap C$. Show that $B = C$
Let $A, B,$ and $C$ be the sets such that $A \cup B=A \cup C$ and $A \cap B=A \cap C$. Show that $B = C$
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 $C \subset B$
$\therefore B=C$
Similar Questions
List all the subsets of the set $\{-1,0,1\}.$
List all the subsets of the set $\{-1,0,1\}.$
Make correct statements by filling in the symbols $\subset$ or $ \not\subset $ in the blank spaces:
$\{ x:x$ is a circlein the plane $\} \ldots \{ x:x$ is a circle in thesame plane with radius $1$ unit $\} $
Make correct statements by filling in the symbols $\subset$ or $ \not\subset $ in the blank spaces:
$\{ x:x$ is a circlein the plane $\} \ldots \{ x:x$ is a circle in thesame plane with radius $1$ unit $\} $
List all the elements of the following sers :
$E = \{ x:x$ is a month of a year not having $ 31 $ days ${\rm{ }}\} $
List all the elements of the following sers :
$E = \{ x:x$ is a month of a year not having $ 31 $ days ${\rm{ }}\} $
State whether each of the following set is finite or infinite :
The set of letters in the English alphabet
State whether each of the following set is finite or infinite :
The set of letters in the English alphabet
What universal set $(s)$ would you propose for each of the following :
The set of right triangles
What universal set $(s)$ would you propose for each of the following :
The set of right triangles