Prove that for three sets $$ \mathrm{A}, \mathrm{B} $$ and C . $$ (A \cap B)^{\prime}=A^{\prime} \cup B^{\prime} $$ $$ (A \cup B)^{\prime}=A^{\prime} \cap B^{\prime} $$ Define Cartesian nroduct of two sets.

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De Morgan's Laws state that: 1. The complement of the intersection of two sets is equal to the union of their complements: $$ (A \cap B)^{\prime} = A^{\prime} \cup B^{\prime} $$. 2. The complement of the union of two sets is equal to the intersection of their complements: $$ (A \cup B)^{\prime} = A^{\prime} \cap B^{\prime} $$. The **Cartesian product** of two sets $$ A $$ and $$ B $$ is the set of all ordered pairs where the first element is from $$ A $$ and the second is from $$ B $$. It is denoted as $$ A 	imes B = \{ (a, b) \mid a \in A 	ext{ and } b \in B \} $$.

Prove that for three sets \( \mathrm{A}, \mathrm{B} \) and C . \( (A \cap B)^{\prime}=A^{\prime} \cup B^{\prime} \) \( (A \cup B)^{\prime}=A^{\prime} \cap B^{\prime} \) Define Cartesian nroduct of two sets.

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Prove that for three sets \( \mathrm{A}, \mathrm{B} \) and C . \( (A \cap B)^{\prime}=A^{\prime} \cup B^{\prime} \) \( (A \cup B)^{\prime}=A^{\prime} \cap B^{\prime} \) Define Cartesian nroduct of two sets.

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