### Theory:

Important!
Let us recall difference of two sets.
For any three sets $$A$$, $$B$$ and $$C$$:

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

(ii) $$A - (B \cap C)$$ $$=$$ $$(A-B) \cup (A-C)$$
Example:
1. Let $$A$$ $$=$$ $$\{-5$$, $$-4$$, $$-1$$, $$0$$, $$1$$, $$2\}$$, $$B$$ $$=$$ $$\{-1$$, $$0$$, $$1$$, $$3$$, $$4\}$$ and $$C$$ $$=$$ $$\{-3$$, $$-1$$, $$1$$, $$4$$, $$5\}$$.

Verify that $$A - (B \cup C)$$ $$=$$ $$(A-B) \cap (A-C)$$.

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

$$B \cup C$$ $$=$$ $$\{-1$$, $$0$$, $$1$$, $$3$$, $$4\}$$ $$\cup$$ $$\{-3$$, $$-1$$, $$1$$, $$4$$, $$5\}$$

$$B \cup C$$ $$=$$ $$\{-3$$, $$-1$$, $$0$$, $$1$$, $$3$$, $$4$$, $$5\}$$

$$A - (B \cup C)$$ $$=$$ $$\{-5$$, $$-4$$, $$-1$$, $$0$$, $$1$$, $$2\}$$ $$-$$ $$\{-3$$, $$-1$$, $$0$$, $$1$$, $$3$$, $$4$$, $$5\}$$

$$A - (B \cup C)$$ $$=$$ $$\{-5$$, $$-4$$, $$2\}$$ - - - - - (I)

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

$$A-B$$ $$=$$ $$\{-5$$, $$-4$$, $$-1$$, $$0$$, $$1$$, $$2\}$$ $$-$$ $$\{-1$$, $$0$$, $$1$$, $$3$$, $$4\}$$

$$A-B$$ $$=$$ $$\{-5$$, $$-4$$, $$2\}$$

$$A-C$$ $$=$$ $$\{-5$$, $$-4$$, $$-1$$, $$0$$, $$1$$, $$2\}$$ $$-$$ $$\{-3$$, $$-1$$, $$1$$, $$4$$, $$5\}$$

$$A-C$$ $$=$$ $$\{-5$$, $$-4$$, $$0$$, $$2\}$$

$$(A-B) \cap (A-C)$$ $$=$$ $$\{-5$$, $$-4$$, $$2\}$$ $$\cap$$ $$\{-5$$, $$-4$$, $$0$$, $$2\}$$

$$(A-B) \cap (A-C)$$ $$=$$ $$\{-5$$, $$-4$$, $$2\}$$ - - - - - (II)

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

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

Hence verified.

2. Let $$A$$ $$=$$ $$\{-5$$, $$-4$$, $$-1$$, $$0$$, $$1$$, $$2\}$$, $$B$$ $$=$$ $$\{-1$$, $$0$$, $$1$$, $$3$$, $$4\}$$ and $$C$$ $$=$$ $$\{-3$$, $$-1$$, $$1$$, $$4$$, $$5\}$$.

Verify that $$A - (B \cap C)$$ $$=$$ $$(A-B) \cup (A-C)$$.

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

$$B \cap C$$ $$=$$ $$\{-1$$, $$0$$, $$1$$, $$3$$, $$4\}$$ $$\cap$$ $$\{-3$$, $$-1$$, $$1$$, $$4$$, $$5\}$$

$$B \cap C$$ $$=$$ $$\{$$$$-1$$, $$1$$, $$4$$$$\}$$

$$A - (B \cap C)$$ $$=$$ $$\{-5$$, $$-4$$, $$-1$$, $$0$$, $$1$$, $$2\}$$ $$-$$ $$\{$$$$-1$$, $$1$$, $$4$$$$\}$$

$$A - (B \cap C)$$ $$=$$ $$\{-5$$, $$-4$$, $$0$$, $$2\}$$ - - - - - (I)

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

$$A-B$$ $$=$$ $$\{-5$$, $$-4$$, $$-1$$, $$0$$, $$1$$, $$2\}$$ $$-$$ $$\{-1$$, $$0$$, $$1$$, $$3$$, $$4\}$$

$$A-B$$ $$=$$ $$\{-5$$, $$-4$$, $$2\}$$

$$A-C$$ $$=$$ $$\{-5$$, $$-4$$, $$-1$$, $$0$$, $$1$$, $$2\}$$ $$-$$ $$\{-3$$, $$-1$$, $$1$$, $$4$$, $$5\}$$

$$A-C$$ $$=$$ $$\{-5$$, $$-4$$, $$0$$, $$2\}$$

$$(A-B) \cup (A-C)$$ $$=$$ $$\{-5$$, $$-4$$, $$2\}$$ $$\cup$$ $$\{-5$$, $$-4$$, $$0$$, $$2\}$$

$$(A-B) \cup (A-C)$$ $$=$$ $$\{-5$$, $$-4$$, $$0$$, $$2\}$$ - - - - - (II)

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

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

Hence verified.
Important!
L.H.SLeft Hand Side

R.H.S – Right Hand Side