COT 3100, Spr. 2001 Applications of Discrete Structures Section #1089X - MWF 4th period Dr. Michael P. Frank Lecture #3 Fri., Jan. 12, 2001 1/12/01 Lecture #3

Administrivia Team assignments. Return graded pre-tests. HW#1 will be available@4pm. Due Thu. 5pm! Harris @ career expo Jan. 23+24. Today: Continue §1.1: xor, implies, biconditional, alternative notations, bits and bit strings. Start §1.2: Simple equivalences. 1/12/01 Lecture #3

Review: The Negation Operator The unary negation operator “¬” (NOT) transforms a proposition into its negation. E.g. If p = “I have brown hair.” then ¬p = “I do not have brown hair.” Truth table for NOT: 1/12/01 Lecture #3

Review: Conjunction Truth Table Note that a conjunction p1  p2  …  pn of n propositions will have 2n rows in its truth table. ¬ and  operations together are universal -sufficient to express any truth table! 1/12/01 Lecture #3

Review: Disjunction Truth Table Note that pq means that p is true, or q is true, or both are true! So this operation is also called inclusive or, because it includes the possibility that both p and q are true. “¬” and “” together are also universal. 1/12/01 Lecture #3

The Exclusive Or Operator The binary exclusive-or operator “” (XOR) combines two propositions to form their logical “exclusive or” (exjunction?). p = “I will earn an A in this course,” q = “I will drop this course,” p  q = “I will either earn an A for this course, or I will drop it (but not both!)” 1/12/01 Lecture #3

Exclusive-Or Truth Table Note that pq means that p is true, or q is true, but not both! This operation is called inclusive or, because it excludes the possibility that both p and q are true. “¬” and “” together are not universal. 1/12/01 Lecture #3

Natural Language is Ambiguous Note that English “or” is ambiguous regarding the “both” case! “Pat is a singer or Pat is a writer.” - “Pat is a man or Pat is a woman.” - Need context to disambiguate the meaning! For this class, assume “or” means inclusive.   1/12/01 Lecture #3

The Implication Operator The implication p  q states that p implies q. I.e., If p is true, then q is true; but if p is not true, then q could be either true or false. E.g., p=“You study hard.” q=“You will get a good grade.” p  q = “If you study hard, then you will get a good grade.” (else, it could go either way) 1/12/01 Lecture #3

Implication Truth Table p  q is false only when p is true but q is not true. p  q does not imply that p causes q! p  q does not imply that p or q are ever true! E.g. “(1=0)  pigs can fly” is TRUE! 1/12/01 Lecture #3

Examples of Implications “If this lecture ends, then the sun will rise tomorrow.” True or False? “If Tuesday is a day of the week, then I am a penguin.” True or False? “If 1+1=6, then Bush won the election.” True or False? “If the moon is made of green cheese, then I am richer than Bill Gates.” True or False? 1/12/01 Lecture #3

English Phrases Meaning p  q “p implies q” “if p, then q” “if p, q” “when p, q” “whenever p, q” “q if p” “q when p” “q whenever p” “p only if q” “p is sufficient for q” “q is necessary for p” “q follows from p” “q is implied by p” We will see some equivalent logic expressions later. 1/12/01 Lecture #3

Converse versus Contrapositive Some terminology: The converse of p  q is: q  p. The contrapositive of p  q is: ¬q  ¬ p. One of these two has the same meaning (same truth table) as p  q. Can you figure out which? Contrapositive 1/12/01 Lecture #3

How do we know for sure? Proving the equivalence of p  q and its contrapositive using truth tables: 1/12/01 Lecture #3

The biconditional operator The biconditional p  q states that p is true if and only if (IFF) q is true. p = “Bush won the election.” q = “Bush will be our next president.” p  q = “If and only if Bush won the election, Bush will be our next president.” 1/12/01 Lecture #3

Biconditional Truth Table p  q means that p and q have the same truth value. Note this truth table is the exact opposite of ’s! p  q means ¬(p  q) p  q does not imply p and q are true, or cause each other. 1/12/01 Lecture #3

Boolean Operations Summary We have seen 1 unary operator (4 possible) and 5 binary operators (16 possible). 1/12/01 Lecture #3

Some Alternative Notations 1/12/01 Lecture #3

Bits and Bit Operations A bit is a binary (base 2) digit: 0 or 1. Bits may be used to represent truth values. By convention: 0 represents “true”; 1 represents “false”. Boolean algebra is like ordinary algebra except that variables stand for bits, + means “or”, and multiplication means “and”. 1/12/01 Lecture #3

Bit Strings A Bit string of length n is an ordered series or sequence of n0 bits. By convention, bit strings are written left to right: e.g. the first bit of “1001101010” is 1. When a bit string represents a base-2 number, by convention the first bit is the most significant bit. Ex. 11012=8+4+1=13. 1/12/01 Lecture #3

Bitwise Operations Boolean operations can be extended to operate on bit strings as well as single bits. E.g.: 01 1011 0110 11 0001 1101 11 1011 1111 Bit-wise OR 01 0001 0100 Bit-wise AND 10 1010 1011 Bit-wise XOR 1/12/01 Lecture #3

End of §1.1 You have learned about: Propositions: What they are. Propositional logic operators’ Symbolic notations. English equivalents. Logical meaning. Truth tables. Atomic vs. compound propositions. Alternative notations. Bits and bit-strings. Next section: §1.2 Propositional equivalences. How to prove them. 1/12/01 Lecture #3

Propositional Equivalence (§1.2) Two syntactically (i.e., textually) different compound propositions may be the semantically identical (i.e., have the same meaning). We call them equivalent. Learn: Various equivalence rules or laws. How to prove equivalences using symbolic derivations. 1/12/01 Lecture #3

Tautologies and Contradictions A tautology is a compound proposition that is true no matter what the truth values of its atomic propositions are! Ex. p  p [What is its truth table?] A contradiction is a comp. prop. that is false no matter what! Ex. p  p [Truth table?] Other comp. props. are contingencies. 1/12/01 Lecture #3

Logical Equivalence Compound proposition p is logically equivalent to compound proposition q, written pq, IFF the compound proposition pq is a tautology. Compound propositions p and q are logically equivalent to each other IFF p and q contain the same truth values as each other in all rows of their truth tables. 1/12/01 Lecture #3

Proving Equivalence via Truth Tables Ex. Prove that pq  (p  q). F T T T F T T F F T T F T F T T F F F T 1/12/01 Lecture #3

Equivalence Laws These are similar to the arithmetic identities you may have learned in algebra, but for propositional equivalences instead. They provide a pattern or template that can be used to match much more complicated propositions and to find equivalences for them. 1/12/01 Lecture #3

Equivalence Laws - Examples Identity: pT  p pF  F Domination: pT  T pF  F Idempotent: pp  p pp  p Double negation: p  p Commutative: pq  qp pq  qp Associative: (pq)r  p(qr) (pq)r  p(qr) 1/12/01 Lecture #3

More Equivalence Laws Distributive: p(qr)  (pq)(pr) p(qr)  (pq)(pr) De Morgan’s: (pq)  p  q (pq)  p  q Trivial tautology/contradiction: p  p  T p  p  F 1/12/01 Lecture #3

Defining Operators via Equivalences Exclusive or: pq  (pq)(pq) pq  (pq)(qp) Implies: pq  p  q Biconditional: pq  (pq)  (qp) pq  (pq) 1/12/01 Lecture #3

TFpqr 1/12/01 Lecture #3