To be more precise, by AND OR and NOT we mean the following functions. It can be easily checked that AND and OR are associative.

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Think of 0 as false and 1 as true.

Boolean algebra and logic gates. To begin we will look at combinations circuits. These are circuits that depend only on the current input value to decide output.

Ch

If more parity bits are used, and if say only one bit is likely to be corrupted in a given word. It may be possible to determine which bit was corrupted, and therefore to correct the error. Suppose we start with 5 data bits and 3 parity bits. The idea is to use 3 parity bits to represent the position of the corrupted bit.

If parity changes, then some odd number of bits have been corrupted. This is called an error detecting code.

Error detecting code: Bits can be corrupted by noise, changing a 0 to a 1 or vice versa. We can add an extra parity bit. So that, for example, every word has an even number of ones called even parity or every word has an odd number of ones which is odd parity.

Given an n bit Gray code we can construct an n plus one bit by reflection.

Gray code is a code in which only 1 bit changes when going from one word to the next.

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BCD is not self complementing

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BCD arithmetic: To represent signed numbers, use signed 10s complement. Add a sign digit of 0000 for non negative numbers...

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To perform addition add BCD digits as if they were binary numbers... if the result is one of the unsigned 4 bit combinations add six to it in binary.

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Signed rs complement: above can be generalized to signed rs complement, where the sign bits are 0 and r minus 1.

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Overflow occurs if and only if adding two number of the same sign gives a number of the other sign.

Addition of signed numbers: Signed 2s complement number can be added in the same way as unsigned 2s complement numbers treating the sign bit like any other... and discarding end carries. We need to watch for overflows which happen when the result lies outside the range of number that can be represented with the bits we have.

Note: To change sign, take the appropriate complement, treating the sign bit like any other. Signed 2s complement is most convenient for computer arithmetic. Note: in n bit signed 2s complement, numbers range from ...

Signed binary numbers

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To change the sign in any of the signed number representations, take the appropriate complement, treating the sign bit like any other. An exception is negative two to the power of n minus one in signed 2s complement its negative is too large.

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1.6 Signed Binary Numbers: Add another bit on the left . If this sign bit is 0, the number is positive. If it s 1, the number is negative.

We can generalize the 2s complement to the rs complement or radix complement. For either type of complement, the complement of the complement is the original number.

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To subtract using the ones complement instead, just add the end carry to the result. If M is greater than or equal to N, this gives the same result as before. If M is less than N, this gives one less than the earlier result i.e. the ones complement of M minus M View page 13 in the textbook for black text on white background...

7 bit adder circuit

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Example of subtraction in binary.

Binary Subtraction using 2s complement: Add the minuend M and the 2s complement of the subtraend N. Discard and end carry. If an end carry is generated, the difference M minus N is nonnegative and the result of the calculation is M minus N in binary. If no end carry is generated the difference M minus N is negative, and the result of the calculation is the 2s complement of N minus M.