Truth table of NAND and NOR can be made from NOT (A AND B) and NOT (A OR B) correspondingly. Exclusive OR (XOR) is a special gate whose output is one only if two inputs aren't equal. Inverse of exclusive OR known as XNOR gate can be a comparator that will generate a 1 output if two inputs are equivalent.
Digital circuits use just one or two kinds of gates for simplicity in fabrication purposes. So one should think in terms of functionally complete set of gates. What does functionally complete set means? A set of gates by that any Boolean function can be realized is known as a functionally complete set. Functionally complete sets are: [AND, NOT], [NOR], [NAND], [OR, NOT].