CS203 Discrete Mathematical Structures Logic Proposition logic

What is logic? The branch of philosophy concerned with analysing the patterns of reasoning by which a conclusion is drawn from a set of premises, without reference to meaning or context (Collins English Dictionary)

Digital Logic

Propositional logic Simple types of statements, called propositions, are treated as atomic building blocks for more complex statements 1) Alexandria is a port or a holiday resort. 2) Alexandria is not a port. Therefore, Alexandria is a holiday resort

Basic connectives and truth tables statements (propositions): declarative sentences that are either true or false--but not both. Eg. Ahmed Hassan wrote Gone with the Wind. 2+3=5. not statements: What a beautiful morning! Get up and do your exercises.

Statement (Proposition) A Statement is a sentence that is either True or False 2 + 2 = 4 True Examples: 3 x 3 = 8 False 787009911 is a prime Today is Tuesday. Non-examples: x+y>0 x2+y2=z2 They are true for some values of x and y but are false for some other values of x and y.

Propositional connectives Propositional logic has four connectives Name Read as Symbol negation ‘not’  conjunction ‘and’  disjunction ‘or’  implication ‘if…then…’ 

Logical Operators About a dozen logical operators Similar to algebraic operators + * - / In the following examples, p = “Today is Friday” q = “Today is my birthday”

Logical operators: Not A “not” operation switches (negates) the truth value Symbol:  or ~ p = “Today is not Friday” p p T F

Logical operators: And An “and” operation is true if both operands are true Symbol:  It’s like the ‘A’ in And pq = “Today is Friday and today is my birthday” p q pq T F

Logical operators: Or An “or” operation is true if either operands are true Symbol:  pq = “Today is Friday or today is my birthday (or possibly both)” p q pq T F

Logical operators: Conditional A conditional means “if p then q” Symbol:  pq = “If today is Friday, then today is my birthday” p→q=¬pq p q pq T F

Logical operators: Bi-conditional A bi-conditional means “p if and only if q” Symbol:  Alternatively, it means “(if p then q) and (if q then p)” Note that a bi-conditional has the opposite truth values of the exclusive or p q pq T F

Boolean operators summary not and or xor conditional Bi-conditional p q p q pq pq pq pq pq T F Learn what they mean, don’t just memorize the table!

Precedence of operators Just as in algebra, operators have precedence 4+3*2 = 4+(3*2), not (4+3)*2 Precedence order (from highest to lowest): ¬   → ↔ The first three are the most important This means that p  q  ¬r → s ↔ t yields: (p  (q  (¬r)) → s) ↔ (t) Not is always performed before any other operation

Example s: Aya goes out for a walk. t: The moon is out. u: It is snowing. : If the moon is out and it is not snowing, then Aya goes out for a walk. If it is snowing and the moon is not out, then Aya will not go out for a walk.

Translating English Sentences p = “It is below freezing” q = “It is snowing” It is below freezing and it is snowing It is below freezing but not snowing It is not below freezing and it is not snowing It is either snowing or below freezing (or both) If it is below freezing, it is also snowing It is either below freezing or it is snowing, but it is not snowing if it is below freezing That it is below freezing is necessary and sufficient for it to be snowing pq p¬q ¬p¬q pq p→q ((pq)¬(pq))(p→¬q) p↔q

Logical Equivalence p q 1 1 1 1 1

Tables of Logical Equivalences Identity laws Like adding 0 Domination laws Like multiplying by 0 Idempotent laws Delete redundancies Double negation “I don’t like you, not” Commutativity Like “x+y = y+x” Associativity Like “(x+y)+z = y+(x+z)” Distributivity Like “(x+y)z = xz+yz” De Morgan L3

Tables of Logical Equivalences Excluded middle Negating creates opposite Definition of implication in terms of Not and Or

Fundamentals of Logic A compound statement is called a tautology(T0) if it is true for all truth value assignments for its component statements. If a compound statement is false for all such assignments, then it is called a contradiction(F0). : tautology : contradiction

Propositional Logic - 2 more defn… A tautology is a proposition that’s always TRUE. A contradiction is a proposition that’s always FALSE. p p p  p p  p T F T F

Tautology example Demonstrate that [¬p (p q )]q is a tautology in two ways: Using a truth table – show that [¬p (p q )]q is always true Using a proof (will get to this later).

Tautology by truth table p q ¬p p q ¬p (p q ) [¬p (p q )]q T F

Tautology by truth table p q ¬p p q ¬p (p q ) [¬p (p q )]q T F

Tautology by truth table p q ¬p p q ¬p (p q ) [¬p (p q )]q T F

Tautology by truth table p q ¬p p q ¬p (p q ) [¬p (p q )]q T F

Tautology by truth table p q ¬p p q ¬p (p q ) [¬p (p q )]q T F

Tautologies, contradictions and programming Tautologies and contradictions in your code usually correspond to poor programming design. EG: while(x <= 3 || x > 3) x++; if(x > y) if(x == y) return “never got here”;

dual Let s be a statement. If s contains no logical connectives other than and , then the dual of s, denoted sd, is the statement obtained from s by replacing each occurrence of and by and , respectively, and each occurrence of T and F by F and T, respectively. Eg. The dual of is

The Principle of Duality Theorem () Let s and t be statements. If , then . First Substitution Rule (replace each p by another statement q) Ex. is a tautology. Replace each occurrence of p by is also a tautology.

Converse The converse of a logical implication is the reversal of the implication. I.e. the converse of p q is q p. EG: The converse of “If Donald is a duck then Donald is a bird.” is “If Donald is a bird then Donald is a duck.” p q and q p are not logically equivalent.

Converse p q p q q p (p q)  (q p)

Converse p q p q q p (p q)  (q p) T F

Logical Non-Equivalence of Conditional and Converse p q p q q p (p q)  (q p) T F

Logical Non-Equivalence of Conditional and Converse p q p q q p (p q)  (q p) T F

Logical Non-Equivalence of Conditional and Converse p q p q q p (p q)  (q p) T F

Fundamentals of Logic Compare the efficiency of two program segments. z:=4; for i:=1 to 10 do begin x:=z-1; y:=z+3*i; if ((x>0) and (y>0)) then writeln(‘The value of the sum x+y is’, x+y) end . if x>0 then if y>0 then … Number of comparisons? 20 vs. 10+3=13 logically equivalent

Derivational Proof Techniques When compound propositions involve more and more atomic components, the size of the truth table for the compound propositions increases Q1: How many rows are required to construct the truth-table of: ( (q(pr ))  ((sr)t) )  (qr ) Q2: How many rows are required to construct the truth-table of a proposition involving n atomic components?

Derivational Proof Techniques A1: 32 rows, each additional variable doubles the number of rows A2: In general, 2n rows Therefore, as compound propositions grow in complexity, truth tables become more and more unwieldy. Checking for tautologies/logical equivalences of complex propositions can become a chore, especially if the problem is obvious.

Derivational Proof Techniques EG: consider the compound proposition (p p )  ((sr)t) )  (qr ) Q: Why is this a tautology? L3

Derivational Proof Techniques A: Part of it is a tautology (p p ) and the disjunction of True with any other compound proposition is still True: (p p )  ((sr)t ))  (qr ) T  ((sr)t ))  (qr ) T Derivational techniques formalize the intuition of this example. L3

Propositional Logic - an unfamous  if NOT (blue AND NOT red) OR red then… (p  q)  q  p  q (p  q)  q  (p  q)  q DeMorgan’s (p  q)  q Double negation p  (q  q) Associativity p  q Idempotent

Tautology by proof [¬p (p q )]q L3

Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive

Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive  [ F  (¬p q)]q ULE L3

Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive  [ F  (¬p q)]q ULE  [¬p q ]q Identity L3

Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive  [ F  (¬p q)]q ULE  [¬p q ]q Identity  ¬ [¬p q ]  q ULE L3

Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive  [ F  (¬p q)]q ULE  [¬p q ]q Identity  ¬ [¬p q ]  q ULE  [¬(¬p) ¬q ]  q DeMorgan L3

Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive  [ F  (¬p q)]q ULE  [¬p q ]q Identity  ¬ [¬p q ]  q ULE  [¬(¬p) ¬q ]  q DeMorgan  [p  ¬q ]  q Double Negation L3

Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive  [ F  (¬p q)]q ULE  [¬p q ]q Identity  ¬ [¬p q ]  q ULE  [¬(¬p) ¬q ]  q DeMorgan  [p  ¬q ]  q Double Negation  p  [¬q q ] Associative L3

Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive  [ F  (¬p q)]q ULE  [¬p q ]q Identity  ¬ [¬p q ]  q ULE  [¬(¬p) ¬q ]  q DeMorgan  [p  ¬q ]  q Double Negation  p  [¬q q ] Associative  p  [q ¬q ] Commutative L3

Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive  [ F  (¬p q)]q ULE  [¬p q ]q Identity  ¬ [¬p q ]  q ULE  [¬(¬p) ¬q ]  q DeMorgan  [p  ¬q ]  q Double Negation  p  [¬q q ] Associative  p  [q ¬q ] Commutative  p  T ULE L3

Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive  [ F  (¬p q)]q ULE  [¬p q ]q Identity  ¬ [¬p q ]  q ULE  [¬(¬p) ¬q ]  q DeMorgan  [p  ¬q ]  q Double Negation  p  [¬q q ] Associative  p  [q ¬q ] Commutative  p  T ULE  T Domination L3

Writing Logical Formula for a Truth Table Digital logic: Given a digital circuit, we can construct the truth table. Now, suppose we are given only the truth table (i.e. the specification, e.g. the specification of the majority function), how can we construct a digital circuit (i.e. formula) using only simple gates (such as AND, OR, NOT) that has the same function?

Examples “I don’t study well and fail” is logically equivalent to “If I study well, then I don’t fail” Write a C program that represents the compound proposition (pq)r

Decide whether the following statements are tautologies or contradictions or neither 1. (p  q)  (q  p). 2. (p  q)  (q  ¬p). 3. p  ¬p. 4. (p  ¬q)  (q  ¬p).

Use truth table to find - P  Q  P  R P  -Q  R  P  R  - Q A  B  -C  D  E  F (P ↔ Q)  (P˅ Q) [P ˄ (P  Q)]  Q

Which of the following equivalent to -(P˄Q)