4. Boolean algebra

4.1. Boolean Functions

De/'M'MY'oM. A function / (x₁, x ₂,..., x_M) which takes one of two values 0 or 1 of M variables each of those also assumes one of two values 0 or 1 is called a Boolean function.

Two Boolean functions are said to be equal if for any tuple of values these two functions take equal values.

We have four Boolean functions of one variable and sixteen functions of two

variables. 2 is the number of Boolean functions of M variables.

Let us consider truth tables of functions of one and two variables.

x			^2
0	0	0	1	1
1	0	1	0	1

Functions ^Q (x) and (x) are called constants respectively 0 and 1. The function (x) coincides with a variable x and is called identical, that is ^{(x )}= ^x.

The function (..) takes the values opposite to those of .., and is called a

negation of .. denoted by .. : (..) = .. .

The truth table of a function of two variables is of the form:

.1	. 2	^0		^2	^3	^4	^5	^ 6		^ 8	^9	^10		^12	^13	^14		^15
0	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1		1
0	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1		1
1	0	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1		1
1	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0		1
						..1							.1
Operatio n		0	A		.1	A	.2	e	v	^		.. 2	v	.1	^			1
						. 2							. 2

Th^/e 4.7.2

1. The functions and are constants.

2. The functions ^₁₀ and essentially depend on one variable:

^3 = ^Y1^, ^5 = .2^, ^10 = ..2 ^, ^12 = ..1 .

3. The function = A ..₂ is called conjunction.

4. The function = v ..₂ is called disjunction.

5. The function = ^ ..₂, or ~ ..₂ is called equivalence.

6. The function = @ ..₂ is called the sum taken absolutely 2.

7. The function = ..₂ ^ is called conversion.

8. The function = ^.. ₂ is called implication.

9. The function = ..₂ is called Scheffer's prime.

10. The function = ^ ..₂is called Peirce's arrow.

11. The functions and are called exclusion's functions.