Digital systems are designed by using Logical Gates. These gates are the AND, OR, NOT, NAND, NOR, Ex-OR and Ex-NOR gates. The basic operations are described below with the help of truth tables.

AND: The AND gate is an electronic circuit that gives a high output (1) only if all its inputs are high.  A dot (.) is used to show the AND operation i.e. A.B
OR: The OR gate is an electronic circuit that gives a high output (1) if one or more of its inputs are high.  A plus (+) is used to show the OR operation.
NOT: The NOT gate is an electronic circuit that produces an inverted version of the input at its output. If the input variable is 1, the inverted output is 0.
NAND: This is a NOT-AND gate which is equal to an AND gate followed by a NOT gate.  The outputs of all NAND gates are high if any of the inputs are low.
NOR: This is a NOT-OR gate which is equal to an OR gate followed by a NOT gate.  The outputs of all NOR gates are low if any of the inputs are high.
Ex-OR: The 'Exclusive-OR' gate which will give a high output if either, but not both, of its two inputs are high.
Ex-NOR: The 'Exclusive-NOR' gate circuit does the opposite to the EOR gate. It will give a low output if either, but not both, of its two inputs are high.