Logic and Boolean Algebra

Logic statements  Real life statements  “I use public transportation when the car is broken and I have to go to work”  “All the lights in the house are turned off”  Mathematical statements  “If side AB is equal to side BC and BC is equal to side CA, then the triangle ABC is equilateral”  “If a shape is both a rombus and a rectangle then it is a square”  “This triangle is rectangle”  Are called logic statements  Statements are either TRUE or FALSE

What is logic  Logic is a branch of mathematics that studies statements.  It helps organize mathematicians on proofs and theorems.  Statements “A”, “B” can be combined as follows:  “A or B” is called (inclusive) logical disjunction  “A and B” is called logical conjunction  “not A” is called logical negation or complement  These are called logic operations

Truth Tables ABA and B (A  B) fff ftf tff ttt ABA or B (A  B) fff ftt tft ttt  To see the effect of these operation we draw their truth tables. Anot A (A) ft tf

A few more operations…  Implication  Exclusive disjunction (exclusive or - XOR) ABA then B (A  B) fft ftt tff ttt ABA xor B (A  B) fff ftt tft ttf

Boolean Algebra  Boolean algebra combines two math fields  Algebra (operations, associative laws, etc)  Binary logic: 0,1 (or True, False)  Developed by George Boole, 1815 – 1864  Has three operators  And (ab), OR (a+b), NOT (a’)  Basic binary operation theorems  Identity: a1=a, a+0=a  Null element: a0=0, a+1=1  Idempotency: aa=a, a+a=a  Complements: a+a’=1, aa’=0  Basic algebraic theorems:  Commutativity: ab=ba, a+b=b+a  Associativity: (ab)c=a(bc), (a+b)+c=a+(b+c)  Distributivity: (a+b)c=(ac)+(bc), (ab)+c=(ac)+(bc)  Covering: a(a+b) = a, a+(ab) = a

Examples  Prove that:  (a+b)b’ = ab’  (ab)+b’ = a+b’  (a+b)’ = a’ b’  (ab)’ =a’ + b’

Use of Boolean Algebra: Logic (i.e. digital) circuits

Honestants and Swindlecants  There are two kinds of people (aborigines) on a mysterious island. There are so-called Honestants who speak always the truth, and the others are Swindlecants who always lie  An outside visitor, on his way to the pub, he met three aborigines. One made this statement: "We are all Swindlecants." The second one concluded: "Just one of us is an Honestant." Who are they?  [Hint: use truth tables]

Pandora’s Box  Once upon a time, there was a girl named Pandora, who wanted a bright groom so she made up a few logic problems for the wannabe. This is one of them.  Based upon the inscriptions on the boxes (none or just one of them is true), choose one box where the wedding ring is hidden.  Golden box: "The ring is in this box."  Silver box: "The ring is not in this box."  Lead box: "The ring is not in the golden box."  [Hint: use truth tables]

Pandora’s Box II  And here is the second test. At least one inscription is true and at least one is false. Which means the ring is in the...  Golden box: "The ring is not in the silver box."  Silver box: "The ring is not in this box."  Lead box: "The ring is in this box."  [Hint: use truth tables]