LESSON 2C: TRUTH TABLE - LEARNING OBJECTIVES: Upon completion of the course, the student should be able to: . know the situations in which conjunctions, disjunctions, negations, and conditionals are truth or false; . determine if two given statements are equivalent using truth tables; and . construct a truth table for a given symbolic expression. - CONTENT EXPLORATION: A truth table shows how the truth (T) or falsity (F) of a compound statement depends on the truth or falsity of the simple statement from which it is constructed. In constructing the truth table, start by looking the five logical connectives. 1. Table for Negation (~) ~P This table is easy to understand. If p is true, its negation ~p is false. If p is false, then ~p is true. 2. Table for Conjunction (and), in symbol () Q PAQ P A Q should be true when both P and Q are true, and false otherwise. 3. Table for Disjunction (or), in symbol ( v ) Q PVQ 55 | Page P V Q is true if either P is true or Q is true (or both- remember that we are using "or" in the inclusive sense). It's only false if both P and Q are false. 4. Table for logical implication (then), in symbol (-) Q P-Q To understand why this table is the way it is, consider the following example: "If you get an A, then I'll give you a dollar". The statement will be true if I keep my promise and false if I don't. Suppose it's true that you get an A and it is true that I give you a dollar. Since I kept my promise, the implication is true. This corresponds to the first line in the table. Suppose it's true that you get an A but it is false that I give you a dollar. Since I didn't keep my promise, the implication is false. This corresponds to the second line in the table. What if it is false that you can get an A? Whether or not I give you a dollar, I haven't broken my promise. Thus, the implication can't be false, so (since this is a two-valued logic) it must be true. This explains the last two lines of the table. 5. Table for Biconditional (if and only if), in symbol () P Q P+ +Q means that P and Q are equivalent. So the double implication is true if P and Q are both true or if P and Q are both false; otherwise, the double implication is also means true.When you are listing the possibilities, you should assign truth values to the component statements in a systematic way to avoid duplication or omission. The easiest approach is to use lexicographic ordering. Thus, for a compound statement with three components P, Q, and R, 2" = 23 = 8, the list would look like this. Q R To summarize the logical connectives above, here is the final table to construct truth tables easier and more precise. Logical and implies biconditional Connectives not or ( ~ ) P ( A ) (V) ( - ) ( + ) Q Examples: Generate the correct formula for the statements below then construct its respective truth table. 1. The art show was enjoyable but the room was hot. P: The art show was enjoyable. Q: The room was hot. Answer. P A Q P Q P Q TRUE TRUE TRUE TRUE TRUE TRUE FALSE TRUE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE 57 | Page Note: You can use letter only (T or F) instead of using complete word (TRUE or FALSE). 2. If the tire is flat then I will have to remove it and take it to the gas station. P: The tire is flat. Q: I have to remove the tire. R: I have to take the tire to the gas station. Answer. P - (Q AR) Q R QR P -> (Q R) TRUE TRUE TRUE TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE FALSE TRUE FALSE FALSE TRUE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE 3. If the boss doesn't like me or thinks I am lazy then she will not give me a raise and I will have to find another apartment. A: The boss likes me. B: The boss thinks I am lazy. C: The boss will give me a raise. D: I will have to find another apartment. Answer. (~ A VB) - (~ CAD) A B D ~A ~Av B C ~CAD (AVB) -> ( ~CD) TRUE TRUE TRUE TRUE FALSE TRUE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE FALSE FALSE TRUE TRUE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE FALSE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE4. The sun is hot but it is not humid. P: The sun is hot. Q: It is humid. Answer: P A~ Q P Q PA~Q TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE 5. If John doesn't pass then he will lose his scholarship and drop out of school. P: John does pass. Q: John will lose his scholarship. R: John will drop out of school. Answer: ~ P - (Q AR) P Q R QAR ~p ~P .> (Q * R) TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE FALSE FALSE TRUE TRUE FALSE TRUE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE 6. If it rains and you don't open your umbrella then you will get wet. A: It rains. B: You open your umbrella. C. You get wet. Answer: (A A~ B) - C A B C ~B As~B (A * ~B) -> C TRUE TRUE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE FALSE TRUE TRUE FALSE FALSE TRUE TRUE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE TRUE FALSE FALSE TRUE TRUE FALSE TRUE FALSE FALSE FALSE TRUE FALSE TRUE 59 | Page 7. If your car won't start or you don't wake up on time then you will miss your interview and you will not get the new job. A: Your car will start. B: You wake up on time. C: You will miss your interview. D: You will get the new job Answer: ( ~ AV~ B) - (CA~ D) A B D ~A "Av ~B CA ~D (Av ~B) -> (CA ~D TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE RUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE RUE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE FALSE TRUE TRUE FALSE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE TRUE TRUE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE TRUE FALSE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE TRUE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE FALSE FALSE 8. If you elect Mary then Mary will make sure that the federal budget will be balanced, partisan wrangling in Washington will cease, and there will be no cuts in social security benefits. E: You elect Mary. B: Mary will make sure that the federal budget will be balanced. C: Mary will make sure partisan wrangling in Washington will cease. S: Mary will make sure that there will be no cuts in social security benefits. Answer: E - (BACAS)- EXERCISE 2C. 1: TRUTH TABLE Direction: Generate a truth table for the following statements. Input your answer in short coupon bond and send your output (computerize) in an email add lailani091094@gmail.com. Deadline of submission is on 7:00 am of the next meeting. 1. AV ~B 2. (AAB) V (C -~ A) 3. P - (~ QVR) 4. (~ AVB) -~ C 5. (AA~ B) - (CVD)