Lecture 02: Propositional Logic

Lecture 02: Propositional Logic CSCI 358 Discrete Mathematics, Spring 2016 Hua Wang, Ph.D. Department of Electrical Engineering and Computer Science January 19, 2015

Propositional logic Propositional logic The simplest logic Definition: A proposition is a statement that is either true or false. Examples: Mines is located Golden, Colorado. (T) 5 + 2 = 8. (F) It is raining today. (either T or F) 2 of 16 CS 441 Discrete mathematics for CS

Composite statements Composite statements More complex propositional statements can be build from elementary statements using logical connectives. Example: Proposition A: It rains outside Proposition B: We will see a movie A new (combined) proposition: If it rains outside then we will see a movie 3 of 16 CS 441 Discrete mathematics for CS

CS 441 Discrete mathematics for CS Composite statements Composite statements More complex propositional statements can be build from elementary statements using logical connectives. Logical connectives: Negation Conjunction Disjunction Exclusive or Implication Biconditional 3 of 16

Negation Negation Definition: Let p be a proposition. The statement "It is not the case that p." is another proposition, called the negation of p. The negation of p is denoted by p and read as "not p." Example: Golden is located at west of Denver. It is not the case that Golden is located at west of Denver. Other examples: 5 + 2 8. 10 is not a prime number. It is not the case that buses stop running at 9:00pm. 4 of 16 CS 441 Discrete mathematics for CS

Negation Negation A truth table displays the relationships between truth values (T or F) of different propositions. p p T F F T Rows: all possible values of elementary propositions: 4 of 16 CS 441 Discrete mathematics for CS

Conjunction Conjunction Definition: Let p and q be propositions. The proposition "p and q" denoted by p q, is true when both p and q are true and is false otherwise. The proposition p q is called the conjunction of p and q. Examples: Colorado School of Mines is located at west of Denver and 5 + 2 = 8. It is raining today and 2 is a prime number. 2 is a prime number and 5 + 2 8. 13 is a perfect square and 9 is a prime. 5 of 16 CS 441 Discrete mathematics for CS

Disjunction Disjunction Definition: Let p and q be propositions. The proposition "p or q" denoted by p q, is false when both p and q are false and is true otherwise. The proposition p q is called the disjunction of p and q. Examples: Colorado School of Mines is located at west of Denver or 5 + 2 = 8. It is raining today or 2 is a prime number. 2 is a prime number or 5 + 2 8. 13 is a perfect square or 9 is a prime. 6 of 16 CS 441 Discrete mathematics for CS

Truth tables of conjunction Truth tables and disjunction Conjunction and disjunction Four different combinations of values for p and q p q p q p q T T T F F T F F Rows: all possible combinations of values for elementary propositions: 2 n values 7 of 16 CS 441 Discrete mathematics for CS

Truth tables of conjunction Truth tables and disjunction Conjunction and disjunction Four different combinations of values for p and q p q p q p q T T T T F F F T F F F F NB: p q (the or is used inclusively, i.e., p q is true when either p or q or both are true). 7 of 16 CS 441 Discrete mathematics for CS

Truth tables of conjunction Truth tables and disjunction Conjunction and disjunction Four different combinations of values for p and q p q p q p q T T T T T F F T F T F T F F F F NB: p q (the or is used inclusively, i.e., p q is true when either p or q or both are true). 7 of 16 CS 441 Discrete mathematics for CS

Exclusive or Exclusive or Definition: Let p and q be propositions. The proposition "p exclusive or q" denoted by p q, is true when exactly one of p and q is true and it is false otherwise. p q p q T T F T F T F T T F F F 8 of 16 CS 441 Discrete mathematics for CS

CS 441 Discrete mathematics for CS Implication Implication Definition: Let p and q be propositions. The proposition "p implies q" denoted by p q is called implication. It is false when p is true and q is false and is true otherwise. In p q, p is called the hypothesis and q is called the conclusion. 9 of 16 p q p q T T T T F F F T T F F T

Implication Implication p q is read in a variety of equivalent ways: if p then q p only if q p is sufficient for q q whenever p Examples: if Paris is a city of U.S. then 2 is a prime. If F then T? 9 of 16 CS 441 Discrete mathematics for CS

CS 441 Discrete mathematics for CS Implication Implication p q is read in a variety of equivalent ways: if p then q p only if q p is sufficient for q q whenever p Examples: if Paris is a city of U.S. then 2 is a prime. T if today is Tuesday then 2 * 3 = 8. What is the truth value? 9 of 16

Implication Implication p q is read in a variety of equivalent ways: if p then q p only if q p is sufficient for q q whenever p Examples: if Paris is a city of U.S. then 2 is a prime. T if today is Tuesday then 2 * 3 = 8. If T then F 9 of 16 CS 441 Discrete mathematics for CS

CS 441 Discrete mathematics for CS Implication Implication p q is read in a variety of equivalent ways: if p then q p only if q p is sufficient for q q whenever p Examples: if Paris is a city of U.S. in 2013 then 2 is a prime. T if today is Thursday then 2 * 3 = 8. F 9 of 16

Implication Implication The converse of p q is q p. The contrapositive of p q is q p The inverse of p q is p q 9 of 16 CS 441 Discrete mathematics for CS

Implication Implication The converse of p q is q p. The contrapositive of p q is q p The inverse of p q is p q Examples: If it snows, the traffic moves slowly. p: it snows q: traffic moves slowly. p q The converse: q p 9 of 16 CS 441 Discrete mathematics for CS

CS 441 Discrete mathematics for CS Implication Implication The contrapositive of p q is q p The inverse of p q is p q Examples: If it snows, the traffic moves slowly. The contrapositive: If the traffic does not move slowly then it does not snow. q p The inverse: p q 9 of 16

Biconditional Biconditional Definition: Let p and q be propositions. The biconditional p q (read p if and only if q), is true when p and q have the same truth values and is false otherwise. p q p q T T T T F F F T F F F T Note: two truth values always agree. 10 of 16 CS 441 Discrete mathematics for CS

Propositional logic: review Propositional logic: 11 of 16

Propositional logic: review Propositional logic: a formal language for representing knowledge and for making logical inferences. 11 of 16

Propositional logic: review Propositional logic: a formal language for representing knowledge and for making logical inferences. A proposition 11 of 16

Propositional logic: review Propositional logic: a formal language for representing knowledge and for making logical inferences. A proposition is a statement that is either True or False. 11 of 16

Propositional logic: review Propositional logic: a formal language for representing knowledge and for making logical inferences. A proposition is a statement that is either True or False. A compound proposition can be created from other propositions using logical connectives. The truth of a compound proposition is defined by truth values of elementary propositions and the meaning of connectives. The truth table for a compound proposition: table with entries (rows) for all possible combinations of truth values of elementary propositions. 11 of 16

Propositional logic: review compound propositions Given the following two propositions: p: 2 is a prime. True q: 6 is a prime. False Determine the truth value of the following statements: p: p q: p q: p q: p q: p q: q p: 12 of 16

Propositional logic: review binary logic connectives We have defined five binary logical connectives between two propositions p and q, including,,,,. Are there any others that might be useful? Yes. How many binary connectives can there be? A binary logical connective is defined by a truth table with 4 rows. Each of the four rows may be True or False, so there are 2 4 = 16 possible truth tables, and thus 16 possible connectives. Why are some of them not very useful? Six of these are trivial ones that ignore one or both inputs; they correspond to True, False, p, q, p and q. Five of them we have already studied:,,,,. The remaining five are potentially useful: One of them is reverse implication ( instead of ). And the other four are the negations of,,,, the first two of which are sometimes called nand and nor. 13 of 16

Propositional logic: review constructing truth table Constructing the truth table Example: Construct the truth table for Example: constructing the truth table for (p q) ( p q) (p q) ( p q) Rows: all possible combinations of values p q p for p elementary q p q (pq) propositions: ( pq) T T 2 n values T F F T F F 14 of 16 CS 441 Discrete mathematics for CS

Propositional logic: review constructing truth table Constructing the truth table Example: constructing the truth table for (p q) ( p q) p q p p q p q (pq) ( pq) T T F F T F T F Typically the target (unknown) compound proposition and its values Auxiliary compound propositions and their values 14 of 16 CS 441 Discrete mathematics for CS

Propositional logic: review constructing truth table Constructing the truth table Examples: Construct a truth table for Example: constructing the truth table for (p q) ( p q) (p q) ( p q) p q p p q p q (pq) ( pq) T T F T F F T F F F T F F T T T T T F F T T F F 14 of 16 CS 441 Discrete mathematics for CS

Precedence of logic connectives Precedence of Logical Operators Precedence of Logical Operators. Operator 15 of 16 Precedence 1 2 3 4 5 We can construct compound propositions using the negation defined so far. We will generally use parentheses to specify t in a compound proposition are to be applied. For instance, of p q and r. However, to reduce the number of parent operator is applied before all other logical operators. This m of p and q, namely, ( p) q, not the negation of the conjun Another general rule of precedence is that the conjunct the disjunction operator, so that p q r means (p q) this rule may be difficult to remember, we will continue to u the disjunction and conjunction operators is clear. Finally, it is an accepted rule that the conditional and have lower precedence than the conjunction and disjunction p q r is the same as (p q) r. We will use paren ditional operator and biconditional operator is at issue, alth precedence over the biconditional operator. Table 8 displays operators,,,,, and.

Precedence of logic connectives Precedence of Logical Operators. Operator 15 of 16 Precedence 1 2 3 4 5 Precedence of Logical Operators Examples We can construct compound propositions using the negation defined so far. iswe applied will generally before use other parentheses connectives: to specify t in a compound proposition are to be applied. For instance, of p q and r. However, to reduce the number of parent operator is applied before all other logical operators. This m of p and q, namely, is applied ( p) before q, : not the negation of the conjun Another general rule of precedence is that the conjunct the disjunction operator, so that p q r means (p q) this rule may be difficult to remember, we will continue to u and have lower precedence than the disjunction and conjunction operators is clear. and : Finally, it is an accepted rule that the conditional and have lower precedence than the conjunction and disjunction p q r is the same as (p q) r. We will use paren ditional operator and biconditional operator is at issue, alth precedence over the biconditional operator. Table 8 displays operators,,,,, and.

Precedence of logic connectives Precedence of Logical Operators. Operator 15 of 16 Precedence 1 2 3 4 5 Precedence of Logical Operators Examples We can construct compound propositions using the negation defined so far. iswe applied will generally before use other parentheses connectives: to specify t in a compound proposition are to be applied. For instance, p q is same as p ( q). of p q and r. However, to reduce the number of parent operator is applied before all other logical operators. This m of p and q, namely, is applied ( p) before q, : not the negation of the conjun Another general rule of precedence is that the conjunct the disjunction operator, so that p q r means (p q) this rule may be difficult to remember, we will continue to u and have lower precedence than the disjunction and conjunction operators is clear. and : Finally, it is an accepted rule that the conditional and have lower precedence than the conjunction and disjunction p q r is the same as (p q) r. We will use paren ditional operator and biconditional operator is at issue, alth precedence over the biconditional operator. Table 8 displays operators,,,,, and.

Precedence of logic connectives Precedence of Logical Operators. Operator 15 of 16 Precedence 1 2 3 4 5 Precedence of Logical Operators Examples We can construct compound propositions using the negation defined so far. iswe applied will generally before use other parentheses connectives: to specify t in a compound proposition are to be applied. For instance, p q is same as p ( q). of p q and r. However, to reduce the number of parent operator is applied before all other logical operators. This m of p and q, namely, is applied ( p) before q, : not the negation of the conjun Another general p q rule r isofsame precedence as (p isq) that r. the conjunct the disjunction operator, so that p q r means (p q) this rule may be difficult to remember, we will continue to u and have lower precedence than the disjunction and conjunction operators is clear. and : Finally, it is an accepted rule that the conditional and have lower precedence than the conjunction and disjunction p q r is the same as (p q) r. We will use paren ditional operator and biconditional operator is at issue, alth precedence over the biconditional operator. Table 8 displays operators,,,,, and.

Precedence of logic connectives Precedence of Logical Operators. Operator 15 of 16 Precedence 1 2 3 4 5 Precedence of Logical Operators Examples We can construct compound propositions using the negation defined so far. iswe applied will generally before use other parentheses connectives: to specify t in a compound proposition are to be applied. For instance, p q is same as p ( q). of p q and r. However, to reduce the number of parent operator is applied before all other logical operators. This m of p and q, namely, is applied ( p) before q, : not the negation of the conjun Another general p q rule r isofsame precedence as (p isq) that r. the conjunct the disjunction operator, so that p q r means (p q) this rule may be difficult to remember, we will continue to u and have lower precedence than the disjunction and conjunction operators is clear. and : Finally, it is an accepted rule that the conditional and have lower precedence p q than r is the same conjunction as (p q) and disjunction r. p q r is the same as (p q) r. We will use paren ditional operator and biconditional operator is at issue, alth precedence over the biconditional operator. Table 8 displays operators,,,,, and.

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