Definition: A proposition is a sentence that is either true or false, but not both.

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Jewel Ball
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1 Math 3336 Section 1.1 Propositional Logic What is a proposition? Definition: A proposition is a sentence that is either true or false, but not both. Examples of Propositions: a. Austin is the capital of exas. b. exas is the largest state of the United States. c = 1 Examples that are NO Propositions: a. Watch out! b. What time is it? c. xx + 3 = 5 ry this one: Classify the following as a true statement, false statement, or neither. a. xx > 5 b. Washington DC is the capital of the United States. c. Moon is made of cheese. d. Keep calm. e. Nike manufactures the world s best running shoes. f. his sentence is false. Letters are used to denote propositions: pp, qq, rr, ss he truth value of a proposition that is always true denoted by, the truth value of a proposition that is always false denoted by. New propositions (compound propositions) can be formed from existing propositions using logical operators. Definition: Let pp be a proposition. he negation of pp, denoted by pp, the statement It is not the case that pp. a. Proposition: A triangle has three sides. Negation: It is not the case that triangle has three sides. Negation in simple English: A triangle does not have three sides. b. Proposition: All fish can swim. Negation: It is not the case that all fish can swim. Negation in simple English: Some fish cannot swim. Page 1 of 8

2 ry this one What is the negation of each of these propositions? a. Proposition: = 5 Negation: b. Proposition: Mike has more than 100 friends on acebook. Negation: A proposition and its negation have OPPOSIE truth values! Construct a truth table for the negation of pp. pp pp Definition: Let pp and qq be propositions. he conjunction of pp and qq, denoted by pp qq, is the proposition pp and qq. he conjunction is true when BOH pp and qq are true and is false otherwise. ry this one: Construct a truth table for the conjunction. pp qq pp qq Page 2 of 8

3 ry this one: ind the conjunction of the following propositions and determine its truth value. a. pp: All birds can fly. qq: = 5 Conjunction: Definition: Let pp and qq be propositions. he disjunction of pp and qq, denoted by pp qq, is the proposition pp or qq. he disjunction is false when BOH pp and qq are false and is true otherwise. ry this one: Construct the truth table for the disjunction. pp qq pp qq ry this one: ind the disjunction of the following propositions and determine its truth value. a. pp: riangles are square. qq: Circles are round. Disjunction: b. pp: 2 = 5 qq: = 7 Disjunction: Page 3 of 8

4 Definition: Let pp and qq be propositions. he exclusive of pp and qq, denoted by pp qq, is the proposition pp or qq, but not both. he exclusive is true when one of pp and qq is true and is false otherwise. Students who have taken calculus or computer science can take this class. Soup or salad comes with this entrée. ry this one: Construct the truth table for the exclusive. pp qq pp qq Definition: Let pp and qq be propositions. he conditional statement (implication) pp qq is the proposition if pp, then qq. he conditional statement pp qq is false then pp is true and qq is false, and true otherwise. In the conditional statement pp qq, pp is called hypothesis and qq is called conclusion. he ruth able for the Conditional Statement pp qq. pp qq pp qq Page 4 of 8

5 Connection between the hypothesis and conclusion is NO necessary. hink: Implication = contract. a. If you get 100% on the final, then you will get an A. b. If the Moon made of cheese, then = 2. Different Ways of Expressing pp qq if p, then q p implies q if p, q p only if q q unless p q when p q if p q whenever p p is sufficient for q q follows from p q is necessary for p a necessary condition for p is q a sufficient condition for q is p Write the statement in the If, then form. a. It is hot whenever it is sunny. b. o get a good grade it is necessary that you study. Definitions: he proposition qq pp is called converse. he proposition pp qq is called inverse. he proposition qq pp is called contrapositive. Page 5 of 8

6 ry this one: Write the converse, inverse, and contrapositive for the following statement. a. If 3 5, then 7 > 7. Converse: Inverse: Contrapositive: b. I come to class whenever there is going to be a quiz. Converse: Inverse: Contrapositive: Definition: Let pp and qq be propositions. he biconditional statement pp qq is the proposition pp if and only if qq. he biconditional statement pp qq is true when p and q have the SAME truth values, and is false otherwise. if and only if = iff You can drive a car if and only if your gas tank is not empty. Page 6 of 8

7 ry this one: he ruth able for the Biconditional Statement pp qq. pp qq pp qq Expressing the Biconditional p is necessary and sufficient for q if p then q, and conversely p iff q ruth ables for Compound Propositions Construction of a truth table: 1. Rows Need a row for every possible combination of values for the atomic propositions. 2. Columns Need a column for the compound proposition (usually at far right) Need a column for the truth value of each expression that occurs in the compound proposition as it is built up. Construct a truth table for (pp qq) (pp qq) Page 7 of 8

8 Equivalent Propositions Definition: wo propositions are equivalent if they always have the same truth value. Show using a truth table that the conditional is equivalent to the contrapositive. Precedence of Logical Operators Operator Precedence pp qq rr is equivalent to If the intended meaning is pp qq rr then parentheses must be used. Page 8 of 8

MA108: Information Packet

MA108: Information Packet Area: Square AA = ss 2 Rectangle Parallelogram Triangle Trapezoid AA = llll AA = bbh AA = 1 bbh 2 AA = 1 h(aa + bb) 2 Circle AA = ππrr 2 Volume: Rectangular Solid VV = llllh Cube VV = ss 3 Pyramid Right