Logic Deals with Statements - Lecture Notes | MATH 1010, Study notes of Mathematics

Download Logic Deals with Statements - Lecture Notes | MATH 1010 and more Study notes Mathematics in PDF only on Docsity! JZsu08 Math 1010 Name:__________________ Logic. Logic deals with statements (or prepositions). The objective of logical analysis of a compound statement is to decide whether it is true or false. DEFINITION: a statement is a declarative sentence that is either true (T) or false (F), but not both. Are the following sentences statements? (Y/N) - “All men are pigs”. ____ - “This is a true statement” ____ - “No way!” ____ - “I don’t have a birthday” ____ - “If you build it, they will come” ____ - “‘Blair Witch Project’ is an excellent movie” ____ - “Guess I’ll be off now” ____ - “It’s the honest truth!” ____ - “ Just kidding ! ” ____ - “Huh ”? ____ Statements can either be simple or compound. Compound statements are made up of simple statements strung together using logical connectives such as ‘and’, ‘or’, ‘not’.The basic logical operators are: operator meaning symbol example negation ‘not’ ~ ~ p conjunction ‘and’ ? p ? q disjunction ‘or’ (inclusive) ? p ? q Note the (fully intentional!) similarities in notation for set operators and logic( , )? ? and connectors . ( , ~ )? ? and The truth value (T or F) of a compound statement depends on the truth value of the simple statements that constitute it. We summarize this dependence in an array called a truth table. The truth table for the three basic operations is defined as follows: p q ~p ~q p ? q p ? q T T F F T T T F F T F T F T T F F T F F T T F F EXAMPLE: Determine the truth value of the clause in a contract: “The car will have a sun roof and power steering, or you’ll receive a 10 % discount” based on the fact that the car is delivered without sun roof but with power steering, yet you do not receive a 10% discount. We start introducing symbols for the simple statements which comprise the clause: s: the car has a sun roof p: the car has power steering d: you receive a 10% discount. The clause can then be written as: (s ? r) ? d where the parentheses are indicated by the comma in the clause. The data are: s ? F , p ? T and d ? F Substituting these truth values in the clause: ( )F T F F F F ? ? ? Conclusion: Based on the data, the clause is false. EXERCISES: Determine the validity of the following statements, based on the given data. Clearly define all simple statements and their symbols, and show your full analysis. 1. A stock market analyst was quoted: “Stocks will go up and Bonds will go down, or bonds will go up and stocks will be down”. Some research showed that for this period, bonds were down and stocks were down, too. 2. A contract for a roofing job contains the phrase: “ 15 year materials will be used throughout or strip ventilation installed, and we will fix damage to gutters resulting from our work”. Upon return from work, you find the yard littered with packing material containing the phrase 10 year warranty, a gutter is badly dented from a ladder, but there is a strip vent on the roof. Should you sue the roofer? 3. Ed’s room mate wants to put a dollar on the following bet: “MTSU will win by more than ten points and Tennessee will lose, or the Titans will win” After the weekend, the results are in: Both Tennessee and the Titans lost, but MTSU won by twenty one points. Who gets the dollar? 11. Consider the statement: "I think, therefore I am ". In words, state the: a. Inverse: ______________________________________________________________. b. Converse: _____________________________________________________________. c. Contrapositive: ________________________________________________________ . d. Fill out the truth table for the above statements: t a ~a ~a t ? a a ? t ~t ? ~a ~a ? ~t T T F F T F T F e. Based on the above truth table, which statements are logically equivalent? ____________ and ______________, also _______________ and _______________ . 12. In mathematics, and indirect proof consists of showing the contrapositive of a statement. Goldbach’s Conjecture: If n is an even integer, then n is the sum of two odd primes. In proving the conjecture indirectly one would start with assuming that ___________________________________________________________________ and go on to show that _____________________________________________________________________. 13. Identify the following arguments by giving their proper names, and determine their validity: a. ______________________ V/NV d. ______________________ V/NV p ? q p ? q p ? q q ? p b. ______________________ V/NV e. ______________________ V/NV p ? q ~q ? ~p p ? q q ? r ? p ? r c. ______________________ V/NV f. ______________________ V/NV p ? q ~p ? ~q p ? q r ? s ? p ? s 14. Determine whether the following arguments are valid or not valid: circle N|NV Support with scheme and give the reason for (non-)validity. Circle all that apply from: VALID: DIRECT|INDIRECT|TRANSITORY reasoning and NON-VALID: INVERSE|CONVERSE|FALSE CHAIN PATTERN fallacy. a. If you eat no meat, you are a vegetarian. You are not a vegetarian. Therefore, you eat meat. b. If the light is red, you have to stop. The light is not red. Therefore, you do not have to stop. c. If I get paid today, I will go tomorrow I did get paid today. Therefore, I am going tomorrow. d. If an animal has protruding front teeth, it's a rodent If an animal has no fur, it is not a rodent Therefore, if an animal has no fur, it has no protruding front teeth. e. If I have time, I will go. I am going. Therefore, I do have the time. f. They say: "(If you have) No pain, (then you’ll have) no gain". I experience pain. Therefore, I will gain. g. If Bama loses, then UT wins. If UT wins, they’ll go to the Cotton Bowl. Therefore, if UT does not go to the Cotton Bowl, Bama won. h. Hot weather favors thunderstorms. Hot weather favors bacterial growth. Bacterial growth causes the milk to go sour. Therefore, thunderstorms cause the milk to go sour. i. If the weather is bad, the game will be canceled. The weather is fair. Therefore, the game will not be canceled. j. If the tree is a cedar, it is an evergreen. If the tree is not an evergreen, it has leaves. Therefore, if the tree has leaves, it is not a cedar. k. Nice clothes make you feel better If you feel better, you'll be healthier Therefore, if you wear nice clothes, you'll be healthier l. Your life expectancy increases if you quit smoking You did not quit smoking Therefore, you can expect to die young _________ ? _________ ? _________ ? _________ ? _________ ? _________ ? _________ ? _________ ? _________ ? _________ ? _________ ? _________ ? V: DIR|INDIR|TRANS NV: INV|CONV|FCP V: DIR|INDIR|TRANS NV: INV|CONV|FCP V: DIR|INDIR|TRANS NV: INV|CONV|FCP V: DIR|INDIR|TRANS NV: INV|CONV|FCP V: DIR|INDIR|TRANS NV: INV|CONV|FCP V: DIR|INDIR|TRANS NV: INV|CONV|FCP V: DIR|INDIR|TRANS NV: INV|CONV|FCP V: DIR|INDIR|TRANS NV: INV|CONV|FCP V: DIR|INDIR|TRANS NV: INV|CONV|FCP V: DIR|INDIR|TRANS NV: INV|CONV|FCP V: DIR|INDIR|TRANS NV: INV|CONV|FCP V: DIR|INDIR|TRANS NV: INV|CONV|FCP