The Laws of Logic Boolean Algebra A State

The Laws of Logic: Boolean Algebra A State High Math Club Presentation START==TRUE

What is Boolean Algebra? • Two values – True and False - Logic + Set Theory – 1 and 0 - Computers + Probability – High and Low – Digital Electronics

AND - Basic • • Notation - AND ^ && (A AND B) returns true iff A and B are both true Numerically: A * B Identity - 1 – 0 AND 1 = 0 – 1 AND 1 = 1 • Annihilator - 0 – 0 AND 0 = 0 – 1 AND 0 = 0

AND - Representations

OR (Inclusive or) - Basic • Notation - OR v || • (A OR B) returns true if either (or both) of A and B are true • Numerically: A + B - A * B – Probability: P(A or B) = P(A)+P(B)-P(A AND B) • “And/Or” Construction • Identity - 0 • Annihilator - 1

OR - Representations

NOT - Basic • Notation NOT ¬ ~ ! • Unary - only takes one argument – Others are called binary • Returns the opposite of its argument • Analogous to negative sign

NOT - Representations

XOR (Exclusive or) - Derived • Notation XOR ^ + • (A XOR B) returns true iff exactly one of A and B is true – Can also be considered “not equals” – “Either/or” construction – A XOR B = (A ^ ~B) v (~A ^ B) • Numerically: A+B (mod 2) – Also A+B – 2*(A*B) – Same formula in probability • A XOR 1 = NOT(A) • A XOR 0 = A

XOR - Representations

Equivalence - Derived • Notation ≡ == • Returns true iff A=B

Material Implication - Derived • Notation x–>y • Linguistically “If X, then Y” – X is called the antecedent; y is called the consequent • False ONLY when x is true and y is false • NOT(X) OR Y – More intuitively NOT(X AND NOT(Y)) • Why is x->y true when x is false? ? ? – This means that “If two is odd, then two is even” is a true statement!

Tautologies: Theorems of Boolean Algebra • Called laws • Proven with either truth tables or derivations – Truth table – true iff last column is all 1 s (i. e. true for all input values) • Example: Proof of X ^ ~X == 0 X ~X X^~X== 0 1 0 1 0 1

De Morgan’s Laws: Distributing a NOT • ¬(x. Vy)=(¬x)^(¬y) • ¬(x^y)=(¬x)V(¬y) • Proofs? X 1 1 0 0 Y 1 0 (¬x)v(¬y) ¬(x^y) 0 0 1 1 1 X Y (¬x)^(¬y) ¬(xvy) 1 1 0 0 0 0 1 1

Important Laws • AND and OR are: – Distributive over each other – Associative – Commutative • • ~~X = X X^~X==0 Xv~X==1 De Morgan’s Laws

A Derivation • (w V x) V (y V z) = ((w V x) V y) V z = (w V (x V y)) V z = (w V (y V x)) V z = ((w V y) V x) V z = (w V y) V (x V z) • Why is this valid?

Applications: Logic + Deduction • Propositional Calculus – gives valid forms of arguments – Arguments are valid iff they are laws of boolean algebra – Tends to use “->” a lot • Examples – (P ^ (P->Q)) -> Q – Modus Ponens – (~Q ^ (P->Q)) -> ~P – Modus Tollens – ((P->Q) ^ (Q->R)) -> (P->R) – Syllogism Hypothetical

Modus Ponens Proof P Q P->Q (P ^ (P->Q)) -> Q 1 1 1 0 0 0 1 1 0 0 1

Modus Tollens Proof P Q P->Q (~Q ^ (P->Q)) -> ~P 1 1 1 0 0 0 1 1 1

Hypothetical Syllogism Proof P 1 1 0 0 Q 1 1 0 0 R 1 0 1 0 P->Q 1 1 0 0 1 1 Q->R P->Q ^ Q->R P->R 1 1 1 0 0 0 1 1 1 1 1

Applications: Computer Curcuits • Everything in a computer is either a 1 or 0 (called a bit, or binary unit) – 1 is high voltage; 0 is low voltage – Why? • Calculations are done with AND, OR, and NOT logic gates – Sound familiar?

How Do Computers Add? • We want to add two numbers, which are sequences of bits – We add one place value (two bits) at a time • Input: The two bits, A and B • Output: sum bit (the actual place value) and carry bit (if we need to carry a 2) • How can we make this circuit?

Adding • Make a truth table and translate it into a circuit diagram A B Sum Carry 1 1 0 1 0 0 1 1 0 0 0

Adding • Sum = A XOR B • Carry = A AND B

What about the carry bit? • We forgot we might need to add a carry bit as well • So we really have three inputs: A, B, and Cin (carry in) • Still two outputs: – Cout (carry out) – Sum

Take Two • How do we make this circuit diagram? A 1 1 0 0 B 1 1 0 0 Cin 1 0 1 0 S 1 0 0 1 1 0 Cout 1 1 1 0 0 0

Full 1 -bit Adder

How about more bits?

How many basic gates? • We use AND, OR, and NOT as basic gates – Wouldn’t it be nice to have ONE basic gate? – Can mass-produce a single circuit; don’t have to worry about which (tiny and impossible to see) circuit is which

NAND – Sheffer Stroke • Notation NAND | ↑ • A NAND B = NOT (A AND B) • Can be used to make AND, OR, and NOT gates – How?

NAND as a Universal Gate • NOT(A) = A NAND A • A AND B = NOT(A NAND B) = (A NAND B) NAND (A NAND B) • A OR B = NOT(A) NAND NOT(B) by Demorgan’s Law = (A NAND A) NAND (B NAND B) • A -> B = NOT(A) OR B = A NAND NOT(B) = A NAND (B NAND B)

NOR – Pierce Arrow • Notation NOR ↓ • A NOR B = NOT(A OR B) • Can be used to make AND, OR, and NOT gates – How?

NOR as a Universal Gate • NOT(A) = A NOR A • A OR B = NOT(A OR B) = (A NOR B) NOR (A NOR B) • A AND B = NOT(A) NOR NOT(B) by Demorgan’s Law = (A NOR A) NOR (B NOR B) • A->B = NOT(A) OR B = (NOT(A) NOR B) NOR (NOT(A) NOR B) = ((A NOR A) NOR B) NOR ((A NOR A) NOR B)

An Everyday Example: Search Engines • Google uses boolean algebra on your search terms – Automatically uses AND (i. e. whitespace = AND) • boolean algebra -> Pages with BOTH ‘boolean’ and ‘algebra’ in them – Use OR for logical OR • Boolean OR algebra -> Pages with EITHER ‘boolean’ or ‘algebra’ in them – Use - for NOT • Boolean –Algebra ->Pages WITH ‘boolean’ but WITHOUT ‘algebra’

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