calculus book - Mathematics and Computer Science

1.1. THE NATURE OF MATHEMATICS 3
Next, consider the following arithmetic/logical operations that are
built from representations of bits:
• (Binary arithmetic) Think of 0 as representing an arbitrary even
integer, and 1 as representing an arbitrary odd integer. That is,
an integer is identified with its remainder on division by 2. The
sum of two odd integers is always even (“1 + 1 = 0”), the product
of an even and an odd integer is always even (“0 · 1 = 0”), and so
forth. If we tabulate the results of addition and multiplication,
we get
+ 0 1
0 0 1
1 1 0
· 0 1
0 0 0
1 0 1
• (Boolean logic) Think of F as representing an arbitrary “false”
assertion (such as “2 + 2 = 5”) and T as representing an arbitrary
“true” sentence (such as “1+1 = 2”). Since “2+2 = 5 or 1+1 = 2,
but not both” is true, we write “F xor T=T”. (“xor” stands for
“exclusive or”: one statement is true, but not both.) Since “2+2 =
5 and 1 + 1 = 2” is false, we write “F and T=F”. The tables below
give the “truth value” of a statement made by conjoining two
statements, according to whether or not each statement is true or
false.
xor F T
F F T
T T F
and F T
F F F
T F T
Each pair of tables encapsulates some structure about bits of data.
The truly mathematical observation is that the entries of the tables
correspond: under the correspondence even-False and odd-True, “addition
(mod 2)” corresponds to “xor”, and “multiplication (mod 2)” corresponds
to “and”. The two pairs of tables above are different implementations
of the same abstract structure, which might even be denoted
∨ • ◦
• • ◦
◦ ◦ •
∧ • ◦
• • •
◦ • ◦