UPSKILL MATH PLUS

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Distributive property of intersection over union
For any three sets $$A$$, $$B$$ and $$C$$: $$A \cap (B \cup C)$$ $$=$$ $$(A \cap B)$$ $$\cup$$ $$(A \cap C)$$
Let $$A$$ $$=$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$$$\}$$, $$B$$ $$=$$ $$\{$$$$n$$, $$o$$, $$p$$, $$q$$, $$r$$$$\}$$ and $$C$$ $$=$$ $$\{$$$$l$$, $$n$$, $$p$$, $$r$$$$\}$$

L.H.S: $$A \cap (B \cup C)$$

$$B \cup C$$ $$=$$ $$\{$$$$n$$, $$o$$, $$p$$, $$q$$, $$r$$$$\}$$ $$\cup$$ $$\{$$$$l$$, $$n$$, $$p$$, $$r$$$$\}$$

$$B \cup C$$ $$=$$ $$\{$$$$l$$, $$n$$, $$o$$, $$p$$, $$q$$, $$r$$$$\}$$

$$A \cap (B \cup C)$$ $$=$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$$$\}$$ $$\cap$$ $$\{$$$$l$$, $$n$$, $$o$$, $$p$$, $$q$$, $$r$$$$\}$$

$$A \cap (B \cup C)$$ $$=$$ $$\{$$$$l$$, $$n$$, $$o$$, $$p$$$$\}$$ - - - - - - - - - (I)

R.H.S: $$(A \cap B)$$ $$\cup$$ $$(A \cap C)$$

$$A \cap B$$ $$=$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$$$\}$$ $$\cap$$ $$\{$$$$n$$, $$o$$, $$p$$, $$q$$, $$r$$$$\}$$

$$A \cap B$$ $$=$$ $$\{$$$$n$$, $$o$$, $$p$$$$\}$$

$$A \cap C$$ $$=$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$$$\}$$ $$\cap$$ $$\{$$$$l$$, $$n$$, $$p$$, $$r$$$$\}$$

$$A \cap C$$ $$=$$ $$\{$$$$l$$, $$n$$, $$p$$$$\}$$

$$(A \cap B)$$ $$\cup$$ $$(A \cap C)$$ $$=$$ $$\{$$$$n$$, $$o$$, $$p$$$$\}$$ $$\cup$$ $$\{$$$$l$$, $$n$$, $$p$$$$\}$$

$$(A \cap B)$$ $$\cup$$ $$(A \cap C)$$ $$=$$ $$\{$$$$l$$, $$n$$, $$o$$, $$p$$$$\}$$ - - - - - - - - - (II)

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

$$A \cap (B \cup C)$$ $$=$$ $$(A \cap B)$$ $$\cup$$ $$(A \cap C)$$

This is called distributive property of intersection over union.
Distributive property of union over intersection
For any three sets $$A$$, $$B$$ and $$C$$: $$A \cup (B \cap C)$$ $$=$$ $$(A \cup B)$$ $$\cap$$ $$(A \cup C)$$
Let $$A$$ $$=$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$$$\}$$, $$B$$ $$=$$ $$\{$$$$n$$, $$o$$, $$p$$, $$q$$, $$r$$$$\}$$ and $$C$$ $$=$$ $$\{$$$$l$$, $$n$$, $$p$$, $$r$$$$\}$$

L.H.S: $$A \cup (B \cap C)$$

$$B \cap C$$ $$=$$ $$\{$$$$n$$, $$o$$, $$p$$, $$q$$, $$r$$$$\}$$ $$\cap$$ $$\{$$$$l$$, $$n$$, $$p$$, $$r$$$$\}$$

$$B \cap C$$ $$=$$ $$\{$$$$n$$, $$p$$, $$r$$$$\}$$

$$A \cup (B \cap C)$$ $$=$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$$$\}$$ $$\cup$$ $$\{$$$$n$$, $$p$$, $$r$$$$\}$$

$$A \cup (B \cap C)$$ $$=$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$, $$r$$$$\}$$ - - - - - - - - - (I)

R.H.S: $$(A \cup B)$$ $$\cap$$ $$(A \cup C)$$

$$A \cup B$$ $$=$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$$$\}$$ $$\cup$$ $$\{$$$$n$$, $$o$$, $$p$$, $$q$$, $$r$$$$\}$$

$$A \cup B$$ $$=$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$, $$q$$, $$r$$$$\}$$

$$A \cup C$$ $$=$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$$$\}$$ $$\cup$$ $$\{$$$$l$$, $$n$$, $$p$$, $$r$$$$\}$$

$$A \cup C$$ $$=$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$, $$r$$$$\}$$

$$(A \cup B)$$ $$\cap$$ $$(A \cup C)$$ $$=$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$, $$q$$, $$r$$$$\}$$ $$\cap$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$, $$r$$$$\}$$

$$(A \cup B)$$ $$\cap$$ $$(A \cup C)$$ $$=$$ $$\{$$$$l$$, $$m$$, $$n$$, $$o$$, $$p$$, $$r$$$$\}$$ - - - - - - - - - (II)

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

$$A \cup (B \cap C)$$ $$=$$ $$(A \cup B)$$ $$\cap$$ $$(A \cup C)$$

This is called distributive property of intersection over union.

Important!
L.H.S – Left Hand Side

R.H.S – Right Hand Side