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According to DeMorgan's theorems, every gate's output may be reversed to produce the same function as the opposing kind of gate (AND vs. OR) with inverted inputs. An extended bar placed over a word serves as a grouping symbol as opposed to the result of inverted terms. This significantly affects the evaluation and reduction of Boolean expressions.
- Mathematician GATE DeMorgan developed two fundamental laws for group complementation in Boolean algebra.
- The complement of a set of words represented by a long bar spanning many variables is referred to as group complementation.
- A gate's primary function, from AND to OR, is reversed when all of its inputs are inverted, and the output is also inverted.
- When all inputs are reversed, negative-OR gates operate exactly like NAND gates and AND gates.
| Table of Content |
Key terms: De Morgan's theorem, gates, boolean algebra, variables, binary algebra.
De Morgan's Theorem
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In digital design, GATE De Morgan's theorems are used to answer Boolean Algebra equations. These theorems, which prove that the complements of the products and sums of all terms are equivalent, are very helpful in boolean algebra. For this usage, two of these theorems are very beneficial.
- GATE DeMorgan's theorems are essential for proving mathematically that NOR and negative-AND gates, as well as negative-OR and NAND gates, are equivalent.
- Using these, boolean algebraic expressions are solved. De-Morgan's theorem's rules are produced using two input variables, x and y
- According to the first theorem, conducting the AND operation on two input variables followed by the NOT operation on the outcome will produce the identical OR operation on the variable's complement.
De-Morgan's First Theorem
The first theorem states that the AND operation's complement result is equivalent to the OR operation's complement, proving (A.B)' = A'+B'.
De-Morgan's Second Theorem
The second theorem states that the OR operation's complement result is equal to its AND operation's complement, proving (A+B)' = A'.B'.
Example 1: (A.B.C)'
(A.B.C)'=A'+B'+C'
Example 2: (A+B+C)'
(A+B+C)'=A'.B'.C
Read More: Digital Electronics and Logic Gates
Boolean Algebra
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In boolean algebra, sometimes referred to as binary algebra or logical algebra, variables have truth values of 1 and 0, respectively. It is crucial to the development of computer languages, set theory, statistics, and digital electronics. It carries out operations for conjunction, disjunction, and negation.
- When logical operators like AND, OR, XOR, and NOT are used, a Boolean expression yields a True or False answer.
- Binary variables and logical operations are the focus of the algebraic discipline known as boolean algebra.
- The definition of a variable is an alphabet for a logical quantity, whereas a function is made up of binary variables, logical operators, constants
- Variables or symbols are what complements and literals are, respectively.
- A truth table is a table that lists every combination of logical variables' potential values.
- It is feasible to turn a Boolean equation into a truth table.
Read More:
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Things to Remember
- De Morgan's theorems are utilised in digital design to solve Boolean Algebra expressions, stating that the complements of all terms are equal.
- Theorem 1 states that the left side of a NAND gate (A and B) has inverted inputs, while the right side represents an OR gate (Blown OR).
- Theorem 2 states that the NOR gate, with inputs A and B, is represented by a left-hand side, while the AND gate, known as a Bubbled AND, has inverted inputs.
- Variables or symbols are what complements and literals are, respectively.
- Binary variables and logical operations are the focus of the algebraic discipline known as boolean algebra.
- A gate's primary function, from AND to OR, is reversed when all of its inputs are inverted, and the output is also inverted.
- A truth table is a table that lists every combination of logical variables' potential values.
Also Read:
Sample Questions
Ques. For what is De Morgan well-known? (2 marks)
Ans. One of the Cambridge mathematicians was De Morgan. He understood that algebra is essentially symbolic in nature. He was well aware that some algebras might be different from regular algebra.
Ques. What effects does De Morgan's theorem have? (2 marks)
Ans. When attempting to demonstrate that the NAND gate is equivalent to an OR gate with inverted inputs and the NOR gate is equivalent to an AND gate with inverted inputs, we can employ De Morgan's theorems. We must first break apart the huge bar expressions before we can lower them.
Ques. What are De Morgan’s theorems? (2 marks)
Ans. De Morgan's theorems are used to answer Boolean algebraic expressions. It is a very effective tool for digital design. According to this theorem, the complements of all the terms' products are equal to the total of their individual complements. Similarly, the products of the complements of each and every term are equal to the complements of the sums of all the terms.
Ques. Write down the truth table and NAND gate's logic symbol. (2 marks)
Ans. A NAND gate, also known as a negative-AND gate, is a logic gate used in digital electronics that only generates a false output if all of its inputs are true. As a result, its output is an AND gate's complement.
Ques. What is a NAND gate? (1 mark)
Ans. The Boolean operator NAND gate, which is identical to NOT AND, returns a value of zero if and only if all of its operands have a value of one.
Ques. The reason NAND & NOR are regarded as universal gates is that they are able to carry out the logical operations related to _______. (1 mark)
(a) AND gate
(b) OR gate
(c) NOT gate
(d) All of the above
Ans. d
Explanation: NAND and NOR gates arе considеrеd univеrsal gatеs in digital logic bеcаusе thеy can bе usеd to perform all thе basic logical opеrations: AND, OR, and NOT. This mеans that one can usе combinations of NAND or NOR gatеs to crеatе circuits that can mimic thе bеhavior of any othеr gatе.
Ques. Which of the boolean expressions listed below violates De-Morgan's theorem? (1 mark)
(a) X+Y = X . Y
(b) X.Y = X + Y
(c) X.Y = X+Y
(d) None of the above
Ans. c
Explanation: This еquation violates Dе-Morgan's thеorеm because it suggеsts that thе complеmеnt of thе product of X and Y is еqual to thе sum of X and Y, which is not truе.
Ques. Being a universal gate, the NOR gate may be changed into an AND gate by flipping its input______. (1 mark)
(a) before getting applied to NOR gate
(b) after getting applied to NOR gate
(c) before getting applied to AND gate
(d) before getting applied to AND gate
Ans. a
Explanation: To changе a NOR gatе into an AND gatе, onе needs to flip its inputs bеforе gеtting applied to thе NOR gate.
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