![Another Example
Prove that ¬[r ∨ (q ∧ (¬r →¬p))] ≡ ¬r ∧ (p∨ ¬q)
¬[r ∨ (q ∧ (¬r → ¬p))]
≡ ¬r ∧ ¬(q ∧ (¬r → ¬p)),
≡ ¬r ∧ ¬(q ∧ (¬¬r ∨ ¬p)),
≡ ¬r ∧ ¬(q ∧ (r ∨¬p)),
≡ ¬r ∧ (¬q ∨ ¬(r ∨ ¬p)),
≡ ¬r ∧ (¬q ∨ (¬r ∧ p)),
≡ (¬r ∧¬q) ∨ (¬r ∧ (¬r ∧ p)),
≡ (¬r ∧¬q) ∨ ((¬r ∧ ¬r) ∧ p),
≡ (¬r ∧¬q) ∨ (¬r ∧ p),
≡ ¬r ∧ (¬q ∨ p),
≡ ¬r ∧ (p ∨¬q),
De Morgan’s law
Conditional rewritten as disjunction
Double negation law
De Morgan’s law
De Morgan’s law, double negation
Distributive law
Associative law
Idempotent law
Distributive law
Commutative law
Discrete Structures(CS 335)
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Discrete Structures lecture 2
- 1. Discrete Structures (CS 335) Lecture 2 Mohsin Raza University Institute of Information Technology PMAS Arid Agriculture University Rawalpindi
- 2. Compound Propositions Producing new propositions from existing propositions. Logical Operators or Connectives 1. Not 2. And ˄ 3. Or ˅ 4. Exclusive or 5. Implication 6. Biconditional Discrete Structures(CS 335) 2
- 3. Compound Propositions Negation of a proposition Let p be a proposition. The negation of p, denoted by p (also denoted by ~p), is the statement “It is not the case that p”. The proposition p is read as “not p”. The truth values of the negation of p, p, is the opposite of the truth value of p. Discrete Structures(CS 335) 3
- 4. Examples 1. Find the negation of the following proposition p : Today is Friday. The negation is p : It is not the case that today is Friday. This negation can be more simply expressed by p : Today is not Friday. Discrete Structures(CS 335) 4
- 5. Examples 2. Write the negation of “6 is negative”. The negation is “It is not the case that 6 is negative”. or “6 is nonnegative”. Discrete Structures(CS 335) 5
- 6. Truth Table (NOT) • Unary Operator, Symbol: p p true false false true Discrete Structures(CS 335) 6
- 7. Conjunction (AND) Definition Let p and q be propositions. The conjunction of p and q, denoted by p˄q, is the proposition “p and q”. The conjunction p˄q is true when p and q are both true and is false otherwise. Discrete Structures(CS 335) 7
- 8. Examples 1. Find the conjunction of the propositions p and q, where p : Today is Friday. q : It is raining today. The conjunction is p˄q : Today is Friday and it is raining today. Discrete Structures(CS 335) 8
- 9. Truth Table (AND) • Binary Operator, Symbol: p q pq true true true true false false false true false false false false Discrete Structures(CS 335) 9
- 10. Disjunction (OR) Definition Let p and q be propositions. The disjunction of p and q, denoted by p˅q, is the proposition “p or q”. The disjunction p˅q is false when both p and q are false and is true otherwise. Discrete Structures(CS 335) 10
- 11. Examples 1. Find the disjunction of the propositions p and q, where p : Today is Friday. q : It is raining today. The disjunction is p˅q : Today is Friday or it is raining today. Discrete Structures(CS 335) 11
- 12. Truth Table (OR) • Binary Operator, Symbol: p q pq true true true true false true false true true false false false Discrete Structures(CS 335) 12
- 13. Exclusive OR (XOR) Definition Let p and q be propositions. The exclusive or of p and q, denoted by pq, is the proposition “pq”. The exclusive or, p q, is true when exactly one of p and q is true and is false otherwise. Discrete Structures(CS 335) 13
- 14. Examples 1. Find the exclusive or of the propositions p and q, where p : Atif will pass the course CSC102. q : Atif will fail the course CSC102. The exclusive or is pq : Atif will pass or fail the course CSC102. Discrete Structures(CS 335) 14
- 15. Truth Table (XOR) • Binary Operator, Symbol: p q pq true true false true false true false true true false false false Discrete Structures(CS 335) 15
- 16. Examples (OR vs XOR) The following proposition uses the (English) connective “or”. Determine from the context whether “or” is intended to be used in the inclusive or exclusive sense. 1. “Nabeel has one or two brothers”. A person cannot have both one and two brothers. Therefore, “or” is used in the exclusive sense. Discrete Structures(CS 335) 16
- 17. Examples (OR vs XOR) 2. To register for BSC you must have passed the qualifying exam or be listed as an Math major. Presumably, if you have passed the qualifying exam and are also listed as an Math major, you can still register for BCS. Therefore, “or” is inclusive. Discrete Structures(CS 335) 17
- 18. Composite Statements Statements and operators can be combined in any way to form new statements. p q p q (p)(q) true true false false false true false false true true false true true false true false false true true true Discrete Structures(CS 335) 18
- 19. Translating English to Logic I did not buy a lottery ticket this week or I bought a lottery ticket and won the million dollar on Friday. Let p and q be two propositions p: I bought a lottery ticket this week. q: I won the million dollar on Friday. In logic form p(pq) Discrete Structures(CS 335) 19
- 21. Conditional Statements Implication Definition: Let p and q be propositions. The conditional statement p q, is the proposition “If p, then q”. The conditional statement p q is false when p is true and q is false and is true otherwise. where p is called hypothesis, antecedent or premise. q is called conclusion or consequence Discrete Structures(CS 335) 21
- 22. Implication (if - then) • Binary Operator, Symbol: P Q PQ true true true true false false false true true false false true Discrete Structures(CS 335) 22
- 23. Conditional Statements Biconditional Statements Definition: Let p and q be propositions. biconditional statement pq, is proposition “p if and only if q”. The the The biconditional (bi-implication) statement p q is true when p and q have same truth values and is false otherwise. Discrete Structures(CS 335) 23
- 24. Biconditional (if and only if) • Binary Operator, Symbol: P Q PQ true true true true false false false true false false false true Discrete Structures(CS 335) 24
- 25. Composite Statements • Statements and operators can be combined in any way to form new statements. P Q P Q (P)(Q) true true false false false true false false true true false true true false true false false true true true Discrete Structures(CS 335) 25
- 26. Equivalent Statements P Q (PQ) (P)(Q) (PQ)(P)(Q) true true false false true true false true true true false true true true true false false true true true • Two statements are called logically equivalent if and only if (iff) they have identical truth tables • The statements (PQ) and (P)(Q) are logically equivalent, because (PQ)(P)(Q) is always true. Discrete Structures(CS 335) 26
- 27. Tautologies and Contradictions • Tautology is a statement that is always true regardless of the truth values of the individual logical variables • Examples: • R(R) • (PQ) (P)(Q) • If S T is a tautology, we write S T. • If S T is a tautology, we write S T. Discrete Structures(CS 335) 27
- 28. Tautologies and Contradictions • A Contradiction is a statement that is always false regardless of the truth values of the individual logical variables Examples • R(R) • ((PQ)(P)(Q)) • The negation of any tautology is a contradiction, and the negation of any contradiction is a tautology. Discrete Structures(CS 335) 28
- 29. Exercises •We already know the following tautology: •(PQ) (P)(Q) •Nice home exercise: •Show that (PQ) (P)(Q). •These two tautologies are known as De Morgan’s laws. Discrete Structures(CS 335) 29
- 30. Logical Equivalence Definition Two proposition form are called logically equivalent if and only if they have identical truth values for each possible substitution of propositions for their proposition variable. The logical equivalence of proposition forms P and Q is written P≡Q Discrete Structures(CS 335) 30
- 31. Equivalence of Two Compound Propositions P and Q 1. Construct the truth table for P. 2. Construct the truth table for Q using the same proposition variables for identical component propositions. 3. Check each combination of truth values of the proposition variables to see whether the truth value of P is the same as the truth value of Q. Discrete Structures(CS 335) 31
- 32. Equivalence Check a. If in each row the truth value of P is the same as the truth value of Q, then P and Q are logically equivalent. b. If in some row P has a different truth value from Q, then P and Q are not logically equivalent. Discrete Structures(CS 335) 32
- 33. Example • Prove that ¬ (¬p)≡ p Solution p T F ¬p F T ¬ (¬p) T F As you can see the corresponding truth values of p and ¬ (¬p) are same, hence equivalence is justified. Discrete Structures(CS 335) 33
- 34. Example Show that the proposition forms ¬(pq) and ¬p ¬q are NOT logically equivalent. p T T F F q T F T F ¬p F F T T ¬q F T F T (pq) ¬(pq) ¬p¬q T F F F F T T T F F F T Here the corresponding truth values differ and hence equivalence does not hold Discrete Structures(CS 335) 34
- 35. De Morgan’s laws De Morgan’s laws state that: The negation of an and proposition is logically equivalent to the or proposition in which each component is negated. The negation of an or proposition is logically equivalent to the and proposition in which each component is negated. Discrete Structures(CS 335) 35
- 36. Symbolically (De Morgan’s Laws) 1. ¬(pq) ≡ ¬p¬q 2. ¬(pq) ≡ ¬p¬q Discrete Structures(CS 335) 36
- 37. Applying De-Morgan’s Law Question: Negate the following compound Propositions 1. John is six feet tall and he weights at least 200 pounds. 2. The bus was late or Tom’s watch was slow. Discrete Structures(CS 335) 37
- 38. Solution a) John is not six feet tall or he weighs less than 200 pounds. b) The bus was not late and Tom’s watch was not slow. Discrete Structures(CS 335) 38
- 39. Inequalities and De Morgan’s Laws Question Use De Morgan’s laws to write the negation of -1< x 4 Solution: The given proposition is equivalent to -1 < x and x 4, By De Morgan’s laws, the negation is -1 ≥ x or x > 4. Discrete Structures(CS 335) 39
- 40. Tautology and Contradiction Definition A tautology is a proposition form that is always true regardless of the truth values of the individual propositions substituted for its proposition variables. A proposition whose form is a tautology is called a tautological proposition. Definition A contradiction is a proposition form that is always false regardless of the truth values of the individual propositions substituted for its proposition variables. A proposition whose form is a contradiction is called a contradictory proposition. Discrete Structures(CS 335) 40
- 41. Example Show that the proposition form p¬p is a tautology and the proposition form p¬p is a contradiction. p ¬p p ¬p p ¬p T F T F F T T F Exercise: If t is a tautology and c contradiction, show that pt≡p and pc≡c? Discrete Structures(CS 335) is 41
- 42. Laws of Logic 1. Commutative laws pq ≡ qp ; pq ≡ qp 2. Associative laws p (q r) ≡ (p q) r ; p(q r) ≡ (pq)r 3. Distributive laws p (q r ) ≡ (p q) (p r) p (q r) ≡ (p q) (p r) Discrete Structures(CS 335) 42
- 43. Laws of Logic 4. Identity laws p t ≡ p ; pc ≡ p 5. Negation laws p¬p ≡ t ; p ¬p ≡ c 6. Double negation law ¬(¬p) ≡ p 7. Idempotent laws p p ≡ p ; pp ≡ p Discrete Structures(CS 335) 43
- 44. Laws of Logic 8. Universal bound laws pt≡t ;pc≡ c 9. Absorption laws p (pq) ≡ p ; p (p q) ≡ p 10. Negation of t and c ¬t ≡ c ; ¬c ≡ t Discrete Structures(CS 335) 44
- 45. Exercise Using laws of logic, show that ⌐(⌐p q) (p q) ≡ p. Solution Take ⌐(⌐p q) (p q) ≡ (⌐(⌐p) ⌐q) (p q), (by De Morgan’s laws) ≡ (p ⌐q) (p q), ≡ p (⌐q q), (by double negative law) (by distributive law) Discrete Structures(CS 335) 45
- 46. contd… ≡ p (q ⌐q), (by the commutative law) ≡ p c, (by the negation law) ≡ p, (by the identity law) Skill in simplifying proposition forms is useful in constructing logically efficient computer programs and in designing digital circuits. Discrete Structures(CS 335) 46
- 47. Another Example Prove that ¬[r ∨ (q ∧ (¬r →¬p))] ≡ ¬r ∧ (p∨ ¬q) ¬[r ∨ (q ∧ (¬r → ¬p))] ≡ ¬r ∧ ¬(q ∧ (¬r → ¬p)), ≡ ¬r ∧ ¬(q ∧ (¬¬r ∨ ¬p)), ≡ ¬r ∧ ¬(q ∧ (r ∨¬p)), ≡ ¬r ∧ (¬q ∨ ¬(r ∨ ¬p)), ≡ ¬r ∧ (¬q ∨ (¬r ∧ p)), ≡ (¬r ∧¬q) ∨ (¬r ∧ (¬r ∧ p)), ≡ (¬r ∧¬q) ∨ ((¬r ∧ ¬r) ∧ p), ≡ (¬r ∧¬q) ∨ (¬r ∧ p), ≡ ¬r ∧ (¬q ∨ p), ≡ ¬r ∧ (p ∨¬q), De Morgan’s law Conditional rewritten as disjunction Double negation law De Morgan’s law De Morgan’s law, double negation Distributive law Associative law Idempotent law Distributive law Commutative law Discrete Structures(CS 335) 47
- 48. Lecture Summery • Logical Connectives • Truth Tables • Compound propositions • Translating English to logic and logic to English. • Logical Equivalence • Equivalence Check • Tautologies and Contradictions • Laws of Logic • Simplification ofDiscrete Structures(CS 335) Compound Propositions 48