If $U = \{1,2,3,4,5,6,7,8,9,10\}$, $A = \{1,2,3,4,5,6\}$ and $B = \{2,3,4\}$, then prove that (i) $(A \cup B)' = A' \cap B'$ (ii) $(A \cap B)' = A' \cup B'$

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>

> [!TLDR]
> Compute each side of both De Morgan identities directly from the given sets and show they are equal.
>
> Given: $U=\{1,2,3,4,5,6,7,8,9,10\}$, $A=\{1,2,3,4,5,6\}$, $B=\{2,3,4\}$.
>
> First find the basic sets:
> $A' = U - A = \{7,8,9,10\}$
> $B' = U - B = \{1,5,6,7,8,9,10\}$
>
> **(i) Prove $(A\cup B)' = A' \cap B'$**
>
> Since $B \subset A$, $A \cup B = A = \{1,2,3,4,5,6\}$.
> $(A\cup B)' = A' = \{7,8,9,10\}$
>
> $A' \cap B' = \{7,8,9,10\} \cap \{1,5,6,7,8,9,10\} = \{7,8,9,10\}$
>
> Both sides equal $\{7,8,9,10\}$, so $(A\cup B)' = A'\cap B'$ ✓
>
> **(ii) Prove $(A\cap B)' = A' \cup B'$**
>
> Since $B \subset A$, $A \cap B = B = \{2,3,4\}$.
> $(A\cap B)' = U - \{2,3,4\} = \{1,5,6,7,8,9,10\}$
> …

Operation	What it asks	Bucket contains
Union ( $A \cup B$ )	In either?	Everything from $A$ or $B$ (or both)
Intersection ( $A \cap B$ )	In both?	Only the overlap
Difference ( $A \setminus B$ )	In $A$ but not $B$ ?	Just the part of $A$ that doesn't touch $B$
Complement ( $A^c$ )	Not in $A$ ?	Everything outside $A$ (inside $U$ )
Symmetric Difference ( $A \triangle B$ )	In exactly one?	The two crescent-shaped parts, excluding the overlap

Q.If $U = \{1,2,3,4,5,6,7,8,9,10\}$ , $A = \{1,2,3,4,5,6\}$ and $B = \{2,3,4\}$ , then prove that

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Q.If U={1,2,3,4,5,6,7,8,9,10}U = \{1,2,3,4,5,6,7,8,9,10\}U={1,2,3,4,5,6,7,8,9,10}, A={1,2,3,4,5,6}A = \{1,2,3,4,5,6\}A={1,2,3,4,5,6} and B={2,3,4}B = \{2,3,4\}B={2,3,4}, then prove that

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Q.If $U = \{1,2,3,4,5,6,7,8,9,10\}$ , $A = \{1,2,3,4,5,6\}$ and $B = \{2,3,4\}$ , then prove that

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