Discussion of the Operations on Concept Algebra, Set Algebra and Logic Algebra
Conbra
Algebra
May. 26, 06
On the real algebra system the operations must be designed first. These
operations must be to describe the object correctly. For example the
addition of numerical algebra is to get a total from two numbers. The
number is the object of the numerical algebra. As well known the
numerical algebra has four basic binary operations. There are addition,
subtraction, multiplication and division. This is a complete algebra
<1>, because the term at this algebra could be moved between two sides
of the equal sign. For example one equation transformation rule is if A
+ B = C then A = C – B. This equation transformation rule also is a
theorem on numerical algebra.
The concept algebra <2> is the second kind of the complete algebra <1>,
because the term on this algebra could be moved between two sides of
the equal sign. Therefore the equation of concept algebra could be
solved. That has been made the logic reasoning be able to calculate
<3>. The Set algebra and the Logic algebra at book <2> are the
explanation of the concept algebra, so that the set equation and the
logic equation also can be solved systematically. The following table
lists the relation of the basic operations on numerical algebra,
concept algebra, set algebra and logic algebra.
Algebra Kind Addition Subtraction Multiplication Division
Concept Algebra + - *
/
Set Algebra ∪ - ∩ /
Logic Algebra | - & →(back
direction of /)
Numerical Algebra + - *
/
Table 1 Basic Operations List
As we see the basic operations on concept algebra is same as the basic
operations on the numerical algebra, but they have different usage and
different axioms. The operation / is called the division on numerical
algebra, but this operation can be called as “get from” on concept
algebra or set algebra (Part 2 of <2>). The implication → is usury
used by people to be able to explain “if…then…”. These basic
operations on concept algebra, set algebra and logic algebra (Part 3 of
<2>) are difference, but the compound operations on these algebras are
really same. As we see the definition of the compound operations on
concept algebra <2> are as follows
(x @ y) = ((x / y) * (y / x)) GW11
(x # y) = ((x - y) + (y - x)) GW12
(x ⊃ y) = ((x + y) @ x) GW13
(x ⊂ y) = ((x * y) @ x) GW14
x !⊃ y = (x + y) # x GW15
x !⊂ y = (x * y) # x GW16
x ↕ y = (y / x) # x GW17
x !↕ y = (y / x) @ x GW18
The definition of the compound operations on set algebra (Part 2 of
<2>) are as follows
(x @ y) = ((x / y) ∩ (y / x)) GS11
(x # y) = ((x - y) ∪ (y - x)) GS12
(x ⊃ y) = ((x ∪ y) @ x) GS13
(x ⊂ y) = ((x ∩ y) @ x) GS14
x !⊃ y = (x ∪ y) # x GS15
x !⊂ y = (x ∩ y) # x GS16
x ↕ y = (y / x) # x GS17
x !↕ y = (y / x) @ x GS18
The definition of the compound operations on set algebra (Part 3 of
<2>) are as follows
(x @ y) = ((x → y) & (y → x)) GW11
(x # y) = ((x - y) | (y - x)) GW12
(x ⊃ y) = ((x | y) @ x) GW13
(x ⊂ y) = ((x & y) @ x) GW14
x !⊃ y = (x | y) # x GW15
x !⊂ y = (x & y) # x GW16
x ↕ y = (x → y) # x GW17
x !↕ y = (x → y) @ x GW18
These compound operations have the same meaning on these three algebras
listed as table 2.
Compound Operation Meaning
@ Equal, Be the same as
# Equal to the complement of
(correct explanation), Not Equal
⊃ Including
⊂ “is”, Be included by
!⊃ Not include
!⊂ Not included
↕ Independent of, Not related
of
!↕ Not independent of, Related
with
Table 2 Explanation of Compound Operations on Concept Algebra and
so on
The compound operation may have many different explanations that will
be researched.
Reference:
<1> Shilong Wu “Discussing of two kind Complete Algebra”
<2〉Shilong Wu “Concept Algebra”
<3> Shilong Wu “Computational Logic Reasoning”
The references see web: http://conceptalgebras.eponym.com