Question

Prove that A ∩ B = A ∪ B.

Answer 1

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Answer on Question #75939 – Math – Discrete Mathematics

Question

Prove that $A \cap B = A \cup B$.

Solution

$$A \cup B = \{x \mid x \in A \text{ or } x \in B\}$$ $$A \cap B = \{x \mid x \in A \text{ and } x \in B\}$$

Let's show, that the equality $A \cup B = A \cap B$ is not always true, that is, we show that $\exists A$ and $B: A \cup B \neq A \cap B$.

For example, $A = \{1,2,3\}, B = \{3,4,5\}$.

Then $A \cup B = \{1,2,3,4,5\}$ and $A \cap B = \{3\}$. Therefore $A \cup B \neq A \cap B$.

Nevertheless, the equality $A \cup B = A \cap B$ may be true (for example, if $A = B$).