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# How do you represent XOR using NAND gates?

## How do you represent XOR using NAND gates?

An XOR gate is made by connecting four NAND gates as shown below. This construction entails a propagation delay three times that of a single NAND gate. , noting from de Morgan’s Law that a NAND gate is an inverted-input OR gate. This construction uses five gates instead of four.

What is the Boolean algebra of XOR gate?

The Boolean expression for XOR gate cannot determined directly like AND, OR gates. As it is a Hybrid gate, the Boolean expression of output of XOR gate is given by a combining of Multiplication, Addition and inverting of inputs.

What are the rules of XOR gate?

If both inputs are false (0/LOW) or both are true, a false output results. XOR represents the inequality function, i.e., the output is true if the inputs are not alike otherwise the output is false. A way to remember XOR is “must have one or the other but not both”.

### What are the theorems of Boolean algebra?

Laws and Theorems of Boolean Algebra

 6a. X • Y = Y • X Commutative Law 7a. X (Y Z) = (X Y) Z = (X Z) Y = X Y Z Associative Law 7b. X + (Y + Z) = (X + Y) + Z = (X + Z) + Y = X + Y + Z Associative Law 8a. X • (Y + Z) = X Y + X Z Distributive Law 9a. X • Y = X + Y de Morgan’s Theorem

How many NAND gates are required to implement NOT AND OR AND XOR AND XNOR gate?

The number of 2-input NAND gates required to implement a 2-input XOR gate is 4.

Logic Gates Min. number of NOR Gate Min. number of NAND Gate
Half-Subtractor 5 5
Full-Subtractor 9 9

Why XOR gate is called exclusive?

This gate is called XOR or exclusive OR gate because its output is only 1 when its input is exclusively 1. In digital electronics, other logic gates include NOT gates, OR gates, NAND gates, and NOR Gates.

## What is the Boolean expression of a ⊕ B?

Summary of 2-input Logic Gates

Logic Function Boolean Notation
NOT A
NAND A .B
NOR A+B
EX-OR (A.B) + (A.B) or A ⊕ B

What is Boolean expression of XOR and XNOR?

NAND: The Boolean expression for the NAND gate is: $$Y = \overline {A \cdot B}$$. XOR: the Boolean expression for the XOR gate is: $$Y = \bar A \cdot B + A \cdot \bar B$$. XNOR: the Boolean expression for the XNOR gate is: $$Y = A \cdot B + \bar A\bar B$$.

What are the properties of XOR?

XOR properties

• XOR is commutative. This means, a ^ b = b ^ a .
• XOR is associative. This means, a ^( b ^ c ) = ( a ^ b )^ c = ( a ^ c )^ b .
• XOR is self-inverse. This means, any number XOR’ed with itself evaluates to 0. a ^ a = 0.
• 0 is the identity element for XOR. This means, any number XOR’ed with 0 remains unchanged.

### Why XOR is called exclusive OR?

An XOR gate implements an exclusive OR, i.e., a true output result if one, and only one, of the gate inputs, is true. If both the inputs are false (either 0 or LOW) or if both are true, there results in a false output.

How many theorem of Boolean algebra are?

two

Boolean Algebra Theorems
The two important theorems which are extremely used in Boolean algebra are De Morgan’s First law and De Morgan’s second law. These two theorems are used to change the Boolean expression.

What are the 3 laws in Boolean logic?

Description of the Laws of Boolean Algebra
0 . 0 = 0 A 0 AND’ed with itself is always equal to 0. 1 . 1 = 1 A 1 AND’ed with itself is always equal to 1.

## What is the maximum number of NAND gates required to realize an XOR gate?

XOR can be made (and were made in TTL) with four NAND’s. If you follow the normal route of truth table, Karnaugh map and convert to NAND’s it doesn’t easily reduce to four gates, but by inspection of the circuit below this arrangement does actually work correctly.

What is the minimum number of NAND gates required to implement a XOR gate?

4 NAND gates
The logic NAND function can be expressed by the Boolean expression of, A.B. Hence a minimum of 4 NAND gates is required to construct an EX-OR gate.

Why is XOR not a universal gate?

Note that neither the Exclusive-OR gate or the Exclusive-NOR gate can be classed as a universal logic gate as they can not be used on their own or together to produce any other Boolean function.

### Where is XOR gate used?

XOR gates are used in circuits that perform arithmetic operations and calculations, especially in half-adders and adders. The ability of the XOR gate to compare two logic levels and give an output dependent upon the input condition is very useful in many computational circuits.

What is the output of XOR gate with a and B as input?

The XOR and XNOR gates are the hybrids gates. The 2-input OR gate is also known as the Inclusive-OR gate because when both inputs A and B are set to 1, the output comes out 1(high). In the Ex-OR function, the logic output “1” is obtained only when either A=”1″ or B=”1″ but not both together at the same time.

What are 7 logic gates?

There are seven basic logic gates: AND, OR, XOR, NOT, NAND, NOR, and XNOR. The AND gate is so named 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.

## Why XOR is called exclusive or?

How do you find the XOR of two numbers?

To find the XOR of two numbers, follow these instructions:

1. Convert the numbers into the binary representation.
2. Compare the corresponding bits of the two numbers.
3. If only one of the input bits is true (1), the output is true (1). Otherwise, the output is false (0).

Is XOR a Boolean operator?

XOR is one of the sixteen possible binary operations on Boolean operands. That means that it takes 2 inputs (it’s binary) and produces one output (it’s an operation), and the inputs and outputs may only take the values of TRUE or FALSE (it’s Boolean) – see Figure 1.

### How do you write XOR?

The logical operation exclusive disjunction, also called exclusive or (symbolized XOR, EOR, EXOR, ⊻ or ⊕, pronounced either / ks / or /z /), is a type of logical disjunction on two operands that results in a value of true if exactly one of the operands has a value of true.

What are the 6 Boolean laws?

Laws of Boolean Algebra

• Commutative law.
• Associative law.
• Distributive law.
• AND law.
• OR law.
• Inversion law.

What is De Morgan’s theorem?

De Morgan’s Theorem, T12, is a particularly powerful tool in digital design. The theorem explains that the complement of the product of all the terms is equal to the sum of the complement of each term. Likewise, the complement of the sum of all the terms is equal to the product of the complement of each term.

## How do you prove Boolean theorems?

Proof: If A, B and C are three variables, then the grouping of 3 variables with 2 variables in each set will be of 3 types, such as (A + B), (B + C) and(C + A). According to associative law, we need to prove that x = y.

Can we make XOR from NAND?

An XOR gate circuit can be made from four NAND gates. In fact, both NAND and NOR gates are so-called “universal gates” and any logical function can be constructed from either NAND logic or NOR logic alone.

What is the Boolean expression for XOR?

XOR Boolean Expression
It can also be written as: (A + B) (A + B)

### How many NAND gates are required for XOR?

Detailed Solution. The number of 2-input NAND gates required to implement a 2-input XOR gate is 4.

How do you make an XOR gate?

These gates are not basic gates in their own and are constructed by combining with other logic gates. Their Boolean output function is significant enough to be considered as a complete logic gate. The XOR and XNOR gates are the hybrids gates.

2-input Ex-OR gate.

Input Output
A B Y
0 0 0
0 1 1
1 0 1

Why XOR is called Exclusive OR?

When both inputs are equal, either are 1 or both are 0, the output will be logical 0. This is the reason an XOR gate is also called an anti-coincidence gate or inequality detector. This gate is called XOR or exclusive OR gate because its output is only 1 when its input is exclusively 1.

## How XOR is formed?

The Exclusive-OR gate or XOR gate is achieved by combining standard logic gates together. XOR gate is used extensively in error detection circuits, computational logic comparators and arithmetic logic circuits.

How is XOR calculated?

What is XOR function?

The XOR function returns a logical Exclusive Or of all arguments.

### What is the XOR function?

What is XOR operation?

(eXclusive OR) A Boolean logic operation that is widely used in cryptography as well as in generating parity bits for error checking and fault tolerance. XOR compares two input bits and generates one output bit. The logic is simple. If the bits are the same, the result is 0. If the bits are different, the result is 1.

What is the Boolean expression of NAND gate?

Logic NAND Gate Equivalence
Its Boolean expression is denoted by a single dot or full stop symbol, ( . ) with a line or Overline, ( ‾‾ ) over the expression to signify the NOT or logical negation of the NAND gate giving us the Boolean expression of: A.B = Q.

## What are the Laws of Boolean algebra?

What is XOR example?

Examples: 1 XOR 1 = 0. 1 XOR 0 = 1. 0 XOR 1 = 1.

What is meant by XOR operation?

### What does NAND stands for?

What does NAND stand for? Surprisingly, NAND is not an acronym. Instead, the term is short for “NOT AND,” a boolean operator and logic gate. The NAND operator produces a FALSE value only if both values of its two inputs are TRUE.

Why NAND gate is called universal gate?

NAND Gate is called Universal Gates because all the other gates (NOT, AND, OR and NOR)can be created by using this gate.

What is a ‘+ 1 in Boolean law?

Annulment Law – A term AND’ed with a “0” equals 0 or OR’ed with a “1” will equal 1. A . 0 = 0 A variable AND’ed with 0 is always equal to 0. A + 1 = 1 A variable OR’ed with 1 is always equal to 1.

## How many Boolean laws are there?

There are six types of Boolean Laws.

What is XOR method?

Where is NAND used?

The NAND type is found mainly in memory cards, USB flash drives, solid-state drives (those produced since 2009), feature phones, smartphones, and similar products, for general storage and transfer of data.

### Why is it called NAND?

It’s non-volatile, and you’ll find NAND in mass storage devices like USB flash drives and MP3 players. NAND memory is a form of electronically erasable programmable read-only memory (EEPROM), and it takes its name from the NAND logic gate.

Is XOR a universal gate?