We last time we began introducing equivalency laws.

Transcription

1 Monday, January 14 MAD2104 Discrete Math 1 Course website: www/mathfsuedu/~wooland/mad2104 Today we will continue in Course Notes Chapter 22 We last time we began introducing equivalency laws Today we will eventually continue with that, but first we will get sidetracked with a topic for tomorrow s quiz (we began this topic on Friday): Disjunctive Normal Form Along with your quiz in recitation tomorrow, you also have an online quiz, due by 11:59 pm tomorrow See the announcement on Canvas for details

2 Disjunctive Normal Form (DNF) Toward the end of Section 2, Chapter 2 of the course notes, we are introduced to two methods for representing any compound statement in a canonical or standard form These two structures are called the disjunctive normal form (DNF) and the conjunctive normal form In this course, we will omit discussion of conjunctive normal form, and focus on disjunctive normal form

3 (This page was covered on Friday) First, two preliminary definitions: literal, minterm A literal is a logic variable, or the negation of a logic variable Examples: A minterm in two variables (p, q) is a conjunction in which a literal of each variable appears exactly once, with the p literal first Examples and are minterms in two variables, but and are not minterms

4 (This page was covered on Friday) Likewise, a minterm in three variables (p, q, r) is a conjunction in which a literal of each of the three variables appears one time, with the p literal first, the q literal second, and the r literal third Examples p q r and p q r are minterms in three variables Example p q r r p q is a properly formatted minterm in three variables, but is formatted incorrectly Minterms in four variables, five variables, and so on, are defined similarly Important: this context (writing minterms) is the only case in which we will not use grouping symbols when we write a compound statement having more than one binary operator

5 Disjunctive Normal Form Any well-formed symbolic statement s, in n variables, is equivalent to exactly one properly formatted statement formed by the disjunction of minterms in n variables This unique representation is called the Disjunctive Normal Form for the statement s EXAMPLES of Disjunctive Normal Form These examples a presented to show what disjunctive normal form is supposed to look like, in the case of a 2-variable or 3-variable proposition After this, we will show how to use a truth table to derive disjunctive normal form Proposition Disjunctive Normal Form q p ( p q) ( p q) ( p q) q ( p q) ( p q) ( p q) p ( p q) [q (r p)] (q r) (r p) ( r p) q (r p) q ( p q) ( p q) (p q r) (p q r) ( p q r) ( p q r) ( p q r) (p q r) (p q r) ( p q r) ( p q r) ( p q r) (p q r) (p q r) ( p q r) ( p q r) ( p q r)

6 Lexicographic ordering In defining minterm, we specified that within a minterm, the literals must be listed in a certain order If you examine the previous examples of more closely, you will see that in each case the minterms themselves are also arranged in a special way; this ordering of minterms, and of literals within the minterms, is known as lexicographic (or dictionary) ordering For a two-variable proposition, this means: 1 Any minterm with a p must be listed before a minterm with a p 2 If two minterms have the same first term literal, then a minterm with a q is listed before a minterm with a q For a three-variable proposition, this means: 1 Any minterm with a p must be listed before a minterm with a p 2 If two minterms have the same first term, then a minterm with a q is listed before a minterm with a q 3 If two minterms have the same first term and the same second term, then a minterm with an r is listed before a minterm with a r

7 EXAMPLE (lexicographic ordering) The five propositions below are all equivalent, but only one them is formatted properly according to lexicographic order Which one? A (p q r) (p q r) ( p q r) ( p q r) ( p q r) B (q p r) ( p r q) ( r q p) ( r p q) (q p r) C (p q r) (p q r) ( p q r) ( p q r) ( p q r) D (p q r) ( p q r) (p q r) ( p q r) ( p q r) E (q p r) ( r q p) (q p r) ( p r q) ( r p q) Note: the correct answer is the disjunctive normal form for the proposition r ( q p)

8 Use a truth table to derive disjunctive normal form In this course, we will focus on writing the Disjunctive Normal Form for statements in two or three variables This is easy to do, using a truth table The disjunctive normal form for a compound proposition s is derived from the truth table rows in which s is true Each row in which s is true contributes one minterm to the DNF If we set up the truth table in the standard way, then the lexicographic ordering is automatic EXAMPLE We will use a truth table to derive the disjunctive normal form for r ( q p)

9 Now, we return to Friday s topic: EQUIVALENCY LAWS Last week we introduced the following equivalency laws: DeMorgan's Laws Double Negation Law Implication Law Contrapositive Law Negation of Implication Law (p q) p q (p q) p q ( p) p p q p q p q q p (p q) p q

10 There are several other laws for logical equivalence that are analogous to, and have the same names as, familiar properties from algebra: Identity Laws Commutative Laws Associative Laws Distributive Laws

11 EXAMPLE Use equivalency laws to prove p (q r) (r p) (p q) The proof should be formatted as a sequence of equivalencies Do not skip or combine steps in the proof Justify each step

12 Constructing a formal equivalency proof Suppose r, s, are compound propositions When we write a formal proof of the equivalency r s, the proof will be formatted in this way: r p 1 (L 1 ) p 2 (L 2 ) p 3 (L 3 ) p n (L n ) s (L n+1 ) where r p 1 because of equivalency law L 1, p 1 p 2 because of equivalency law L 2, and so on In summary: Your formal proof should be a sequence of equivalencies, each of which is justified by the application of an equivalency law

13 Counterexamples and disproof To disprove a claim of equivalence (that is, to show that a proposed equivalence is false), we find a counterexample In this case, counterexample is an assignment of truth values to variables that causes the two propositions to have opposite truth values EXAMPLE Find a counterexample to show that a conditional p q is not equivalent to its converse q p EXAMPLE Find a counterexample to show that (p q) is not equivalent to p q

14 EXAMPLE Decide whether the following claim is true or false: (p q) r p (q r) A True B False If the claim is true, write the proof If the claim is false, provide a counterexample

15 EXAMPLE Decide whether the following claim is true or false: p (q r) q (p r) A True B False If the claim is true, write the proof If the claim is false, provide a counterexample

16 One last batch The following equivalence laws are especially helpful for piecing together the details needed for some equivalence proofs Tautology law Contradiction law Domination laws Idempotent laws Absorption laws Biconditional equivalency