Computers: Defining Logical Operations
Date: 05/21/98 at 09:40:34 From: Towella Subject: AND, OR, XOR, NAND, NOR, NOT Hi. I was referred to your site and while here tried to find an explanation for the terms: AND, OR, XOR, NAND, NOR, NOT but was unsuccessful. If there is a any way that I could get a brief definition or even an example of what these do, I would be eternally grateful! Thanks, Towella
Date: 05/22/98 at 13:12:29 From: Doctor Jeremiah Subject: Re: AND, OR, XOR, NAND, NOR, NOT Hi Towella: I am a computer engineer and these logical operations are basically all a computer knows how to do at the very lowest level. Everything else is made from these operations. AND is a multi-input operation that returns true when all the inputs are true and returns false the rest of the time. OR is a multi-input operation that returns true when at least one or even more than one input is true and returns false the rest of the time. NOT is a single input operation that returns the opposite of the input. A truth table can be used to show all the combinations of inputs and outputs. For these truth tables A and B are the inputs and Y is the output: AND OR NOT A B | Y A B | Y A | Y -------------+------- -------------+------- -------+------- False False | False False False | False False | True False True | False False True | True True | False True False | False True False | True True True | True True True | True See how AND is only true when all the inputs are true and how OR is true whenever any of the inputs are true? NAND is a NOT on the output of an AND. So it has the exact opposite output as an AND would have. NOR is a NOT on the output of an OR. So it has the exact opposite output as an OR would have. NAND NOR A B | Y A B | Y -------------+------- -------------+------- False False | True False False | True False True | True False True | False True False | True True False | False True True | False True True | False See how the NAND is true unless all the output are true and how NOR is true whenever all the outputs are false. XOR is true whenever an odd number of inputs is true. XNOR is the opposite output from XOR. XOR XNOR A B | Y A B | Y -------------+------- -------------+------- False False | False False False | True False True | True False True | False True False | True True False | False True True | False True True | True There is such a thing as boolean algebra where you can solve equations made up of logical operations. It looks a lot like regular algebra. It is used by computer engineers when designing circuits. AND is expressed the same as multiplication would be. OR is expressed the same as addition would be (with a plus sign). NOT is expressed with an apostrophe. XOR is expressed as a plus sign with a circle around it. And like regular algebra has multiplication set to a higher precedence than addition, boolean algebra does the same with the AND function. So Y = AB' + A'B is: A ANDed with the inverse (NOT) of B and then ORed to the output of the NOT of A ANDed with B. This means that: Y is true if A = true and B' = true. Y would also be true if A' = true and B = true. The reason that there are two independent cases is because of the OR. Remember that A' is the NOT (inverse) of A and B' is the NOT of B. If we take that into account we can change it to: Y is true if A = true and B = false. Y would also be true if A = false and B = true. If you look carefully at the truth table of XOR you will notice that the previous statement IS the definition of XOR. XNOR would be Y = (AB' + A'B)' See the NOT around the whole expression? I hope that helps. If you need more details please write back. -Doctor Jeremiah, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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