Cecily Zamora
De Morgan's theorem, or equivalence law, is a useful tool in digital design. It states that the complement of a product of literals is equivalent to the sum of the complements of the literals, and the complement of a sum of literals is equivalent to the product of the complements of the literals. In the context of XOR and XNOR gates, De Morgan's theorem can be applied to convert between the two. An XOR gate gives a logic 1 output if either, but not both, of its two inputs are logic 1, while an XNOR gate is the opposite, giving a logic 0 output under the same conditions. By inverting the output of an XOR gate, we get an XNOR gate, and vice versa. This can be achieved by adding a NOT function to the output of the XOR gate or by inverting one of its inputs.
| Characteristics | Values |
|---|---|
| XOR gate | Produces a logic 1 output if either, but not both, of its two inputs are logic 1 |
| XNOR gate | Produces a logic 0 output if either, but not both, of its two inputs are logic 1 |
| De Morgan's theorem | The complement of a product of literals is equivalent to the sum of the complements of the literals, and the complement of a sum of literals is equivalent to the product of the complements of the literals |
| Truth table | A table that describes how a gate works |
| NAND gate | A "universal gate" from which any logical function can be constructed |
| NOR gate | A "universal gate" from which any logical function can be constructed |
| Demorgan-style symbols | Alternate symbols for basic gates that can be used to match up bubbles and make logic diagrams |
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What You'll Learn
- De Morgan's theorem states that the complement of a product of literals is equivalent to the sum of the complements of the literals
- The Exclusive OR (XOR) gate gives a logic 1 output if either, but not both, of its two inputs are logic 1
- The Exclusive NOR (XNOR) gate gives a logic 0 output if either, but not both, of its two inputs are logic 1
- The XNOR gate can be obtained by complementing the output of an XOR gate
- The XNOR gate can be obtained by adding a NOT at the output of the XOR gate
De Morgan's theorem states that the complement of a product of literals is equivalent to the sum of the complements of the literals
De Morgan's theorem is a fundamental principle in Boolean algebra, with applications in digital design and electronics. It is comprised of two sets of rules or laws, which can be used to simplify Boolean logic expressions. The first law states that the complement of the union of two sets, A and B, is equal to the intersection of the complements of the sets A and B. This can be expressed as (AUB)' = A'∩B'.
The second law, conversely, states that the complement of the intersection of two sets is the union of their complements. These two laws are collectively referred to as De Morgan's Law. De Morgan's theorem can be used to express logic expressions not originally containing inversion terms in a different way, and to ''break' an inversion, which is the complement of a complex Boolean expression.
In the context of XOR and XNOR gates, De Morgan's theorem can be applied to convert between the two. An XOR gate can be implemented using AND-OR-Invert (AOI) or OR-AND-Invert (OAI) logic, and its truth table describes how the gate works: when the two inputs are different, the output is 1, and when the inputs are the same, the output is 0. The XNOR gate, on the other hand, has an output of 1 when the inputs are the same and an output of 0 when the inputs are different.
By applying De Morgan's theorem, we can invert the output of an XOR gate to obtain an XNOR gate. This can be achieved by adding a NOT gate at the output of the XOR gate or by redrawing the schematic of the XOR gate with a NOT gate. Additionally, an XNOR gate can be converted to an XOR gate by inverting the output or one of the inputs.
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The Exclusive OR (XOR) gate gives a logic 1 output if either, but not both, of its two inputs are logic 1
The Exclusive OR (XOR) gate is a special type of logic gate used in digital electronics to perform the exclusive OR operation. This gate takes two inputs and produces an output depending on the combination of the two inputs applied. The output of the XOR gate is logic 1 if either, but not both, of its two inputs are logic 1. In other words, the output of the XOR gate is true if either input is true, but not both. The output is false if both inputs are false or both inputs are true.
The XOR gate can be implemented using AND-OR-Invert (AOI) or OR-AND-Invert (OAI) logic. An XOR gate using a 2-1 AOI gate and an XOR gate using a 2-2 OAI gate with negated inputs are examples of this. The XOR gate can also be constructed using a combination of basic logic gates. For example, an XOR gate can be constructed using two AND gates, two NOT gates, and one OR gate.
The XOR gate is a fundamental operation in digital logic and has many applications. It is used in comparator circuits to check if two binary values are equal. It is also used in pseudo-random number (PRN) generators, specifically linear-feedback shift registers (LFSR), and in the simplest phase detectors.
The XOR gate can be converted to an XNOR gate by inverting the output or one of the inputs. The XNOR 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. This can be achieved by adding a NOT at the output of the XOR gate or by using De Morgan's Law. De Morgan's Law states that a NOR gate is an inverted-input AND gate, and a NAND gate is an inverted-input OR gate. By replacing four NAND gates with NOR gates, an XNOR gate is created, which can be converted to an XOR gate by inverting the output or one of the inputs.
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The Exclusive NOR (XNOR) gate gives a logic 0 output if either, but not both, of its two inputs are logic 1
The Exclusive NOR (XNOR) 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. In other words, the output of the XNOR gate is logic 1 when both the inputs are the same (either both 1 or both 0) and logic 0 when the inputs are different. Hence, the XNOR gate is used to implement similarity checker circuits.
The XOR gate, on the other hand, acts as a logical either/or. The output is true if either input is true, but not both. The output is false if both inputs are false or both inputs are true. The output is 1 (true or high) if the inputs are different and 0 (false or low) if the inputs are the same.
The XNOR gate can be obtained by inverting the output of an XOR gate. This can be achieved by adding a NOT at the output of the XOR gate. Alternatively, the whole procedure for the XOR gate can be redone, using one less NAND gate. The XNOR gate can also be obtained by replacing the four NAND gates with NOR gates. This results in an XNOR gate, which can be converted to an XOR gate by inverting the output or one of the inputs.
DeMorgan's theorem for basic gates states that all the basic gates can be given DeMorgan symbols. The basic gates, NOT, AND, OR, XOR, and XNOR have alternate DeMorgan-style symbols. By adding or removing a bubble, the gate's identity changes, for example from XOR to XNOR.
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The XNOR gate can be obtained by complementing the output of an XOR gate
The XNOR gate is a logic gate that performs an exclusive NOR operation. It is the complement of the XOR gate and can be obtained by complementing the output of an XOR gate. This can be achieved by adding a NOT at the output of the XOR gate or its circuit.
The XOR gate, or Exclusive OR gate, is a special type of logic gate used in digital electronics that performs the exclusive OR operation. It takes two inputs and produces an output depending on the combination of the two inputs applied. The output of the XOR gate is 1 when the two inputs are different and 0 when the inputs are the same.
The XNOR gate, on the other hand, produces an output of 1 when both inputs are the same and an output of 0 when the inputs are different. This is the opposite, or complement, of the XOR gate's output. Therefore, to convert an XOR gate into an XNOR gate, we need to invert or complement the output of the XOR gate. This can be done by adding a NOT gate at the output of the XOR gate circuit.
By applying De Morgan's theorem, we can also represent this conversion symbolically. The usual symbol for the XOR gate has an output of 1 when A or B is 1, but not both. By adding an output bubble to this symbol, we indicate that the output is now inverted, giving us the XNOR gate.
In terms of circuit design, an XOR gate can be constructed using NAND or NOR gates. Replacing these NAND or NOR gates with their respective complementary gates (NOR or NAND) will result in an XNOR gate. This can also be achieved by inverting the output or one of the inputs of the XOR gate circuit.
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The XNOR gate can be obtained by adding a NOT at the output of the XOR gate
The XNOR gate is the logical complement of the XOR gate. In other words, the XNOR gate is the opposite or inverse of the XOR gate. The XNOR gate can be obtained by adding a NOT gate to the output of the XOR gate. This is because the output of the XOR gate is 1 when A or B, but not both, are 1. By adding a NOT gate to the output, the XNOR gate will have an output of 0 when A or B, but not both, are 1.
The XNOR gate can also be represented by a combination of two NOT gates, two AND gates, and one OR gate. The XNOR gate is also known as the "equivalence gate" because it implements logical equality. The output of the XNOR gate is 1 only when both the input values are the same (i.e. both 0 or both 1). In all other cases, the output is 0.
The XNOR gate can be constructed using De Morgan's Law. De Morgan's Law states that a NOR gate is an inverted-input AND gate, and a NAND gate is an inverted-input OR gate. By replacing four NAND gates with NOR gates, we can obtain an XNOR gate. This XNOR gate can then be converted to an XOR gate by inverting the output or one of the inputs using a fifth NOR gate.
Additionally, an XNOR gate can be constructed using a NAND gate and an OR-AND-invert gate. This construction uses the fact that the XNOR gate is the logical complement of the XOR gate. By using De Morgan's Law, we can transform the expression (A+B).( A+B) to (A.B) + ( A. B) which can be implemented using three gates.
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Frequently asked questions
The Exclusive OR (XOR) gate gives a logic 1 output if either, but not both, of its two inputs are logic 1.
The Exclusive NOR (XNOR) gate gives a logic 0 output if either, but not both, of its two inputs are logic 1.
Yes, De Morgan's Law can be used to turn XOR into XNOR. This can be done by complementing the output of an XOR gate or by inverting the output of an XOR gate.
