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Home » Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. show that B = C.
Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. show that B = C.
CBSE | Class 11 | Maths | Miscellaneous Exercise | NCERT | Sets
Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. show that B = C.

Solution:

As per the question,

A \cup B=A \cup C

And,

A \cap B=A \cap C

We need to show that,

B=C

Let’s suppose,

x \in B

Therefore,

\begin{array}{l} x \in A \cup B \\ x \in A \cup C \end{array}

As a result,

x \in A \text { or } x \in C

When we have x \in A, therefore,

\begin{array}{l} x \in B \\ \therefore x \in A \cap B \\ \text { As, } A \cap B=A \cap C \\ \text { So, } x \in A \cap C \end{array}

\begin{array}{l} \therefore \mathrm{x} \in \mathrm{A} \text { or } \mathrm{x} \in \mathrm{C} \\ \mathrm{x} \in \mathrm{C} \\ \therefore \mathrm{B} \subset \mathrm{C} \end{array}

In the similar way, it can be shown that C \subset B

As a result, \mathrm{B}=\mathrm{C}

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