1 / 27

Propositional Equivalences And Logical Equivalences in Mathematics

Explore the concepts of propositional equivalences and logical equivalences in mathematics, with examples and proofs for better understanding. Learn about tautologies, contradictions, contingencies, and logical equivalence. Discover important logical equivalences and simplification techniques.

hewitth
Download Presentation

Propositional Equivalences And Logical Equivalences in Mathematics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 3: Logic (2) • Propositional Equivalences • Predicates and Quantifiers 310205 Mathematics for Comter I

  2. 3.1. Propositional Equivalences • Categories of compound propositions: • A tautology is a proposition which is always true. • Classic Example: PP • A contradictionis a proposition which is always false. • Classic Example: PP • A contingencyis a proposition which neither a tautology nor a contradiction. • Example: (PQ)R 310205 Mathematics for Comter I

  3. 3.1. Propositional Equivalences • Two propositions P and Q are logically equivalentif P  Q is a tautology. We write P  Q 310205 Mathematics for Comter I

  4. 3.1. Propositional Equivalences • Example: (PQ)(QP) (P  Q) • Proof: • Left side and the right side must have the same truth values, independent of the truth value of the component propositions. • To show a proposition is not a tautology: use an abbreviated truth table • try to find a counter exampleor to disprove the assertion. • search for a case where the proposition is false. 310205 Mathematics for Comter I

  5. 3.1. Propositional Equivalences • Two possible cases: • Case 1: Left side false, right side true. • Case 2: Left side true, right side false. • Case 1: Try left side false, right side true • Left side false: only one of PQ or QPneed be false. 1a. Assume PQ= F. Then P = T, Q = F. But then right side PQ = F. Oops, wrong guess. 1b. Try QP= F. Then Q = T, P = F. But then right side PQ = F. Another wrong guess. *Proof for (PQ)(QP)  (P  Q) 310205 Mathematics for Comter I

  6. 3.1. Propositional Equivalences • Case 2. Try left side true, right side false • If right side is false, P and Q cannot have the same truth value. 2a. Assume P =T, Q = F. Then PQ= F and the conjunction must be false so the left side cannot be true in this case. Another wrong guess. 2b. Assume Q = T, P = F. Then QP= F. Again the left side cannot be true. • We have exhausted all possibilities and not found a counter-example. The two propositions must be logically equivalent. • Note: Given such equivalence, if and only if or iff is also stated as is a necessary and sufficient condition for. *Proof for (PQ)(QP) (P  Q) 310205 Mathematics for Comter I

  7. 3.1. Propositional Equivalences • Some important logical equivalences: 310205 Mathematics for Comter I

  8. 3.1. Propositional Equivalences • Some important logical equivalences: 310205 Mathematics for Comter I

  9. 3.1. Propositional Equivalences • Other logical equivalences: 310205 Mathematics for Comter I

  10. 3.1. Propositional Equivalences • Other logical equivalences: • Equivalent expressions can always be substituted for each other in a more complex expression – useful for simplification. 310205 Mathematics for Comter I

  11. 3.1. Propositional Equivalences • Example: • (P(PQ)) can be simplified by using the following series of logical equivalence: (P(PQ)) P(PQ)) from the second De Morgan’s law P[(P)Q] from the first De Morgan’s law P(PQ) from the double negation law (PP)(PQ] from the distributive law  F(PQ) since PPF PQ from the identity law for F (PQ) from the second De Morgan’s law 310205 Mathematics for Comter I

  12. 3.1. Propositional Equivalences • But Complexity (2n)… REMEMBER! We can always use a truth table to show that the simplified proposition is equivalent to the original proposition. 310205 Mathematics for Comter I

  13. 3.2. Predicates and Quantifiers • 2+1=3: a proposition • x+y=3: propositional functions or predicates • A generalization of propositions • Propositions which contain variables • Predicates become propositions once every variable is bound - by • assigning it a value from the Universe of Discourse U or • quantifying it 310205 Mathematics for Comter I

  14. 3.2.1. Predicates • Example 1: • Let U = Z, the integers = , -2, -1, 0 , 1, 2, 3,  • P(x): x > 0,a predicate or propositional function. • It has no truth value until the variable x is bound. • Examples of propositions where x is assigned a value: • P(-3) is false, i.e. -3 > 0 is false. • P(0) is false. • P(3) is true. • The collection of integers for which P(x) is true are the positive integers. 310205 Mathematics for Comter I

  15. 3.2.1. Predicates • Example 1 (continued): • P(x): x > 0 is the predicate. • P(y)P(0) is not a proposition. The variable y has not been bound. • However, P(3)P(0) is a proposition which is true. • Example 2: • Let R be the three-variable predicate R(x, y, z): x + y = z. Find the truth value of • R(2, -1, 5) • R(3, 4, 7) • R(x, 3, z) 310205 Mathematics for Comter I

  16. 3.2.2. Quantifiers • Specific value vs. Range • What range of values in U for which the bounded propositions are true? • Two possibilities: • Universal: For all values in U • Existential: For some values in U 310205 Mathematics for Comter I

  17. 3.2.2. Quantifiers • Universal • P(x) is true for every x in the universe of discourse. • Notation: universal quantifier xP(x) • For all x, P(x) or • For every x, P(x) • The variable x is bound by the universal quantifier producing a proposition. • Example: • U = { 1,2,3 } x P(x) P(1)P(2)P(3) 310205 Mathematics for Comter I

  18. 3.2.2. Quantifiers • Existential • P(x) is true for some x in the universe of discourse. • Notation: existential quantifierxP(x) • There is an x such that P(x) or • For some x, P(x) • For at least one x, P(x) • I can find an x such that P(x) • Example: • U = { 1,2,3 } x P(x)P(1)P(2)P(3) 310205 Mathematics for Comter I

  19. 3.2.2. Quantifiers REMEMBER! A predicate (propositional function) is not a proposition until all variables have been bound either by quantification or assignment of a value! • Predicate equivalences: • Equivalences involving the negation operator • x P(x)  x P(x) • x P(x)  x P(x) • Distributing a negation operator across a quantifier changes a universal to an existential, and vice versa. 310205 Mathematics for Comter I

  20. 3.2.2. Quantifiers • Multiple Quantifiers: • Read from left to right . . . • Example 1: • Let U = R, the real numbers, P(x,y): xy = 0 xy P(x,y) xy P(x,y) xy P(x,y) xy P(x,y) • The only one that is false is the first one. Why? • Suppose P(x,y) is the predicate x/y=1? 310205 Mathematics for Comter I

  21. 3.2.2. Quantifiers • Dangerous situations: • Commutativity of quantifiers xy P(x, y) yx P(x, y)? YES! xy P(x, y) yx P(x, y)? NO! DIFFERENT MEANING! 310205 Mathematics for Comter I

  22. 3.2.2. Quantifiers • Example of non-commutativity of quantifiers: • Let Q(x, y) denote “x + y = 0.” • Are the truth values of the quantifications yx P(x, y) and xy P(x, y) the same? • The answer is NO since: • yx P(x, y) means “There is a real number y such that for all real numbers x, Q(x, y) is true.” • The statement is false. Why? • x y P(x, y) means “For every real number x there is a real number y such that Q(x, y) is true.” • The statement is true. 310205 Mathematics for Comter I

  23. 3.2.3. Converting from English • (can be very difficult) • Example 1: • Express the statement “If somebody is female and is a parent, then this person is someone’s mother” as a logical expression. • Let F(x): x is female. P(x): x is a parent M(x, y): x is the mother of y. • The statement applies to all people. x ((F(x)  (P(x))  y M(x, y)) 310205 Mathematics for Comter I

  24. 3.2.3. Converting from English • Example 2: • Express the statement “Everyone has exactly one best friend” as a logical expression. • Let B(x, y): y is the best friend of x. • The statement says “exactly one best friend”. This means that if y is the best friend of x, then all other people z other than y can not be the best friend of x. xyz (B(x, y)  ((z  y)  B(x, z))) 310205 Mathematics for Comter I

  25. 3.2.3. Converting from English • Example 3: • Consider the following statements. “All lions are fierce.” “Some lions do not drink coffee.” “Some fierce creatures do not drink coffee.” • The first two are called premises and the third is called the conclusion. The entire set is called an argument. 310205 Mathematics for Comter I

  26. 3.2.3. Converting from English • Example 3 (continued): • We can express these statements as follows. • LetP(x): x is a lion. Q(x): x is fierce. R(x): x drinks coffee. • Then x (P(x)  Q(x)). x (P(x)  R(x)). x (Q(x)  R(x)). • Why can’t we write the second statement as x (P(x)  R(x))? 310205 Mathematics for Comter I

  27. 3.3. Further Readings • Propositional Equivalences • Rosen: Section 1.2. • Predicates and Quantifiers • Rosen: Section 1.3. 310205 Mathematics for Comter I

More Related