Chapter 5: Architecture - Computer and Information Science - CUNY

10 CHAPTER 5. ARCHITECTURE
Exercise 5.2.2 Show that this matrix performs the OR operation.
The NAND gate
is of special importance because every logical gate can be made out of NAND
gates. Let us try to work out which matrix would correspond to NAND. One
way is to sit down and consider for which of the four possible input states of
two bits (00,01,10,11) does NAND output a 1 (answer: 00,01,10) and in which
states does NAND output a 0 (answer: 11). From this we realize that NAND
can be written as
00 01 10 11
[
0 0 0 0 1
NAND =
1 1 1 1 0
]
. (5.35)
Notice that the column names correspond to the inputs and the row names
correspond to the outputs. 1 in the jth column and ith row means that on
entry j the matrix/gate will output i.
There is, however, another way in which one can determine the NAND gate.
The NAND gate is really the AND gate followed by the NOT gate.
In other words, we can perform the NAND operation by first performing the
AND operation and then the NOT operation. In terms of matrices we can write
this as
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
NOT ⋆ AND = ⎣ 0 1 ⎦ ⋆ ⎣ 1 1 1 0 ⎦ = ⎣ 0 0 0 1 ⎦ = NAND. (5.36)
1 0 0 0 0 1 1 1 1 0
Exercise 5.2.3 Find a matrix that corresponds to NOR.
This way of thinking of NAND rings to light a general situation. When we
perform a computation, often we have to carry out one operation followed by
another.