Logic

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Logic

Logic can be viewed as black boxes with binary input (independent variable) and binary output (dependent variable). Also refers to both the study of modes of reasoning and the use of valid reasoning. In the latter sense, logic is used in most intellectual activities.

Logic in computer science has emerged as a discipline. Today, logic is extensively applied in the fields of Artificial Intelligence, and Computer Science, and these fields provide a rich source of problems in formal and informal logic.

Boolean logic, which has been considered is fundamental to computer hardware, particularly, the system's arithmetic and logic structures, relating to operators AND, NOT, and OR.

Logic gates

A logic gate is an elementary building block of a digital circuit. Complex electronic circuits are built using the basic logic gates. At any given moment, every terminal of the logic gate is in one of the two binary conditions low (0) or high (1), represented by different voltage levels.

There are 3 basic logic gates: AND, OR, NOT.

Other 3 gates- NAND, NOR, XOR and XNOR are based on the 3 basic gates.

The AND gate

The AND gate is so called because, if 0 is called "false" and 1 is called "true," the gate acts in the same way as the logical "and" operator. The following illustration and table show the circuit symbol and logic combinations for an AND gate.

The output is "true" when both inputs are "true." Otherwise, the output is "false."

 

The OR gate

The OR gate gets its name from the fact that it behaves after the fashion of the logical "or." The output is "true" if either or both of the inputs are "true." If both inputs are "false," then the output is "false."

The NOT gate

A logical inverter, sometimes called a NOT gate to differentiate it from other types of electronic inverter devices, has only one input. It reverses the logic state (i.e. its input).

As previously considered, the AND, OR and NOT gates’ actions correspond with the AND, OR and NOT operators.

Any more complex functionality can be constructed from the three basic gates by using DeMorgan’s Law:

The NAND gate

The NAND gate operates as an AND gate followed by a NOT gate. It acts in the manner of the logical operation "and" followed by negation. The output is "false" if both inputs are "true." Otherwise, the output is "true". It finds the AND of two values and then finds the opposite of the resulting value.

The NOR gate

The NOR gate is a combination of an OR gate followed by an inverter. Its output is "true" if both inputs are "false." Otherwise, the output is "false". It finds the OR of two values and then finds the complement of the resulting value.

The XOR gate

The XOR (exclusive-OR) gate acts in the same way as the logical "either/or." The output is "true" if either, but not both, of the inputs are "true." The output is "false" if both inputs are "false" or if both inputs are "true." Another way of looking at this circuit is to observe that the output is 1 if the inputs are different, but 0 if the inputs are the same.

XOR gate

A	B	Z
0	0	0
0	1	1
1	0	1
1	1	0

The XNOR gate

The XNOR (exclusive-NOR) gate is a combination of an XOR gate followed by an inverter. Its output is "true" if the inputs are the same, and"false" if the inputs are different. It performs the operation of an XOR gate and then inverts the resulting value.

XNOR gate

A	B	Z
0	0	1
0	1	0
1	0	0
1	1	1

With the combinations of several logic gates, complex operations can be performed by electronic devices. Arrays (arrangement) of logic gates are found in digital integrated circuits (ICs).

As IC technology advances, the required physical volume for each individual logic gate decreases and digital devices of the same or smaller size become capable of performing much-more-complicated operations at an increased speed.

Combination of gates

A goes into the NOT gate and is inverted, after inversion, it goes into the AND gate along with the variables B and C. The final output at the output terminal of the AND gate is BC. More complex circuitry can be developed using the symbolic representation in this same manner.

Questions

1.      Draw the physical representation of the AND, OR, NOT and XNOR logic gates.

2.      Draw the logic circuit and truth table for Z= A.B.C,  W= (P¯.Q) + (R+S¯)