Copyright © 2004-2012 Curt Hill Axiomatic Approach Some needed groundwork.

Copyright © 2004-2012 Curt Hill

Axiomatic Approach

Some needed groundwork


Prior Approaches• We have seen the English analogy

approach to symbolic logic• This uses the common sense

understanding of the connectives to give insight

• We have also seen the truth table approach and Venn diagrams

• Before this we have defined operators in terms of truth tables

• Now a different way


The Axiomatic Approach• The most rigorous approach to Boolean

algebra• Here we give axioms and then reason

from them in an analytic fashion• An axiom is either a basic definition or

self obvious statement that is never proven

• From basic proof or calculational techniques we build up a larger set of theorems

• Once a theorem is proven, then we may use it in exactly the same way as an axiom

Connection• At this point we start with a blank

slate– We know nothing about the operators– We do not even know about the values

• We may use truth tables to convince ourselves that a theorem is true– This is only moral support, it does not

help the proof• What we are doing is pure symbolic

manipulation– It has nothing to do with reality



The nature of our proofs• Usually start with a supposition• Using the properties of our

operators we manipulate the supposition

• Eventually we have manipulated it into an axiom or theorem

• Each step must be obvious and convincing, no hand waving is allowed


Operator properties• These are all properties that an

operator may possess or not• Usually the operator axioms will

state if the operator possesses this property or not

• Sometimes this may be a theorem


Associative Property• An operator is associative if we have

an expression with three variables and two of the same operator and we can parenthesize it any way

• Suppose that @ is an operator• If @ is associative then A @ B @ C

can be rewritten as:– (A @ B) @ C– A @ (B @ C)– Without changing the resulting values


Symmetric• Usually called commutative in

algebra of real numbers, but symmetry here

• An operator is symmetric if it is not sensitive to reversing the order of writing

• If | is symmetric then A | B can be rewritten as B | A


Examples from Real Numbers• Addition and multiplication are

both associative and symmetric (or commutative)– x + y + z = x + (y + z) = (x + y) + z– x + y = y + x

• Subtraction and division are neither


Reflexive• An operator is reflexive if you can

put the same operand on both sides of the operator and the result is true

• The equality operator is reflexive because x=x for any x

• Notice it does not matter what value x has, the statement is true, which means that we could remove x=x and replace with true


Transitive• If a@b and b@c then a@c• Only the comparison operators in

the Algebra of Reals are transitive– Equality, less than, greater than are

all transitive• However, in Boolean Algebra most

operators look like a comparison– They produce a True/False value


Idempotent• An operator is idempotent if you

can put the operator between two of the same variable and get the variable back

• I do not think this one exists in arithmetic operators but we will see it plenty in logical operators

• If @ is idempotent – Then we can always replace A@A is

always equal to just A• Do not confuse with the unit


Distributivity• We say that one operator

distributes over another if the first is outside a paranthesis and the other inside and we can rewrite without the parenthesis

• Multiplication distributes over addition– X*(A+B) = AX + BX

• We will see several distributivities in logic


Precedence or binding• If two operators have the same binding

power or precedence then we can evaluate them in a left to right fashion, otherwise not– Sometimes we will disallow two operators to

be adjacent without parentheses• Recall precedence from real algebra

– Multiplication has higher precedence than addition so:

– A+b*c must have parenthesis in order to evaluate the addition first


Unit• The unit for the operator is that

value that when operated with a variable returns the variable

• For example – Zero is the unit of addition

• A+0 = A– One is the unit of multiplication

• A*1 = A• In algebra we call this the identity


Zero• If an operator has a zero then this

value operated on any value gives back the zero

• Addition has no zero• Multiplication has a zero of zero• Several logic operators have a zero

Copyright © 2004-2012 Curt Hill Axiomatic Approach Some needed groundwork.

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Transcript of Copyright © 2004-2012 Curt Hill Axiomatic Approach Some needed groundwork.