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Important!
Let us recall complement of a set.
Let $$U$$ be the universal set containing finite sets $$A$$ and $$B$$, then:

(i) $$(A \cup B)^{\prime}$$ $$=$$ $$A^{\prime} \cap B^{\prime}$$

(ii) $$(A \cap B)^{\prime}$$ $$=$$ $$A^{\prime} \cup B^{\prime}$$
Example:
1. Let $$U$$ $$=$$ $$\{10$$, $$20$$, $$30$$, $$40$$, $$50$$, $$60$$, $$70$$, $$80$$, $$90\}$$, $$A$$ $$=$$ $$\{10$$, $$30$$, $$50$$, $$60$$, $$70\}$$ and $$B$$ $$=$$ $$\{10$$, $$20$$, $$30$$, $$50$$, $$80\}$$.

Verify that $$(A \cup B)^{\prime}$$ $$=$$ $$A^{\prime} \cap B^{\prime}$$.

L.H.S: $$(A \cup B)^{\prime}$$

$$A \cup B$$ $$=$$ $$\{10$$, $$30$$, $$50$$, $$60$$, $$70\}$$ $$\cup$$ $$\{10$$, $$20$$, $$30$$, $$50$$, $$80\}$$

$$A \cup B$$ $$=$$ $$\{10$$, $$20$$, $$30$$, $$50$$, $$60$$, $$70$$, $$80\}$$

$$(A \cup B)^{\prime}$$ $$=$$ $$\{10$$, $$20$$, $$30$$, $$40$$, $$50$$, $$60$$, $$70$$, $$80$$, $$90\}$$ $$-$$ $$\{10$$, $$20$$, $$30$$, $$50$$, $$60$$, $$70$$, $$80\}$$

$$(A \cup B)^{\prime}$$ $$=$$ $$\{40$$, $$90\}$$ - - - - - - (I)

R.H.S: $$A^{\prime} \cap B^{\prime}$$

$$A^\prime$$ $$=$$ $$\{10$$, $$20$$, $$30$$, $$40$$, $$50$$, $$60$$, $$70$$, $$80$$, $$90\}$$ $$-$$ $$\{10$$, $$30$$, $$50$$, $$60$$, $$70\}$$

$$A^\prime$$ $$=$$ $$\{20$$, $$40$$, $$80$$, $$90\}$$

$$B^\prime$$ $$=$$ $$\{10$$, $$20$$, $$30$$, $$40$$, $$50$$, $$60$$, $$70$$, $$80$$, $$90\}$$ $$-$$ $$\{10$$, $$20$$, $$30$$, $$50$$, $$80\}$$

$$B^\prime$$ $$=$$ $$\{40$$, $$60$$, $$70$$, $$90\}$$

$$A^{\prime} \cap B^{\prime}$$ $$=$$ $$\{20$$, $$40$$, $$80$$, $$90\}$$ $$\cap$$ $$\{40$$, $$60$$, $$70$$, $$90\}$$

$$A^{\prime} \cap B^{\prime}$$ $$=$$ $$\{40$$, $$90\}$$ - - - - - - (II)

From (I) and (II), we see that:

$$(A \cup B)^{\prime}$$ $$=$$ $$A^{\prime} \cap B^{\prime}$$.

Hence verified.

2. Let $$U$$ $$=$$ $$\{10$$, $$20$$, $$30$$, $$40$$, $$50$$, $$60$$, $$70$$, $$80$$, $$90\}$$, $$A$$ $$=$$ $$\{10$$, $$30$$, $$50$$, $$60$$, $$70\}$$ and $$B$$ $$=$$ $$\{10$$, $$20$$, $$30$$, $$50$$, $$80\}$$.

Verify that $$(A \cap B)^{\prime}$$ $$=$$ $$A^{\prime} \cup B^{\prime}$$.

L.H.S: $$(A \cap B)^{\prime}$$

$$A \cap B$$ $$=$$ $$\{10$$, $$30$$, $$50$$, $$60$$, $$70\}$$ $$\cap$$ $$\{10$$, $$20$$, $$30$$, $$50$$, $$80\}$$

$$A \cap B$$ $$=$$ $$\{10$$, $$30$$, $$50\}$$

$$(A \cap B)^{\prime}$$ $$=$$ $$\{10$$, $$20$$, $$30$$, $$40$$, $$50$$, $$60$$, $$70$$, $$80$$, $$90\}$$ $$-$$ $$\{10$$, $$30$$, $$50\}$$

$$(A \cap B)^{\prime}$$ $$=$$ $$\{20$$, $$40$$, $$60$$, $$70$$, $$80$$, $$90\}$$ - - - - - - (I)

R.H.S: $$A^{\prime} \cup B^{\prime}$$

$$A^\prime$$ $$=$$ $$\{10$$, $$20$$, $$30$$, $$40$$, $$50$$, $$60$$, $$70$$, $$80$$, $$90\}$$ $$-$$ $$\{10$$, $$30$$, $$50$$, $$60$$, $$70\}$$

$$A^\prime$$ $$=$$ $$\{20$$, $$40$$, $$80$$, $$90\}$$

$$B^\prime$$ $$=$$ $$\{10$$, $$20$$, $$30$$, $$40$$, $$50$$, $$60$$, $$70$$, $$80$$, $$90\}$$ $$-$$ $$\{10$$, $$20$$, $$30$$, $$50$$, $$80\}$$

$$B^\prime$$ $$=$$ $$\{40$$, $$60$$, $$70$$, $$90\}$$

$$A^{\prime} \cup B^{\prime}$$ $$=$$ $$\{20$$, $$40$$, $$80$$, $$90\}$$ $$\cup$$ $$\{40$$, $$60$$, $$70$$, $$90\}$$

$$A^{\prime} \cup B^{\prime}$$ $$=$$ $$\{20$$, $$40$$, $$60$$, $$70$$, $$80$$, $$90\}$$ - - - - - - (II)

From (I) and (II), we see that:

$$(A \cap B)^{\prime}$$ $$=$$ $$A^{\prime} \cup B^{\prime}$$.

Hence verified.
Important!
L.H.S. – Left Hand Side

R.H.S. – Right Hand Side