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

2. Copyright © 2004-2012 Curt Hill 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

3. Copyright © 2004-2012 Curt Hill 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

4. 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 Copyright © 2004-2012 Curt Hill

5. 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

6. Copyright © 2004-2012 Curt Hill 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

7. Copyright © 2004-2012 Curt Hill 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

8. Copyright © 2004-2012 Curt Hill 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

9. Copyright © 2004-2012 Curt Hill 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

10. Copyright © 2004-2012 Curt Hill 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

11. Copyright © 2004-2012 Curt Hill 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

12. Copyright © 2004-2012 Curt Hill 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

13. Copyright © 2004-2012 Curt Hill 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

14. Copyright © 2004-2012 Curt Hill 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

15. Copyright © 2004-2012 Curt Hill 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

16. Copyright © 2004-2012 Curt Hill 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