PS10 Sets & Logic. Sam Maddy Austin Jacob For each of these statements, list the students for which the statement is true:

Transcription

1 PS10 Sets & Logic Lets check it out: We will be covering A) Propositions, negations, B) Conjunctions, disjunctions, and C) intro to truth tables, D) 3 propositions. hink about this. On Saint Patrick s Day, the students in a class are all encouraged to wear green clothes to school. Sam Maddy Austin Jacob or each of these statements, list the students for which the statement is true: a) I am wearing a green shirt. b) I am not wearing a green shirt. c) I am wearing a green shirt and green pants. d) I am wearing a green shirt or green pants. e) I am wearing a green shirt or green pants, but not both. 1

2 Propositions Propositions are statements which may be true or false. hey are not if a form of a question, but rather an assertion that do not include opinions. Propositions do no have to have the same answer every time its asserted, but rather are indeterminate. he truth value of a proposition is whether it is true or false. [Note that Logic is closely associated with set notation and Venn diagrams.] Examples Which of the following statements are propositions? If they are propositions, are they true, false, or indeterminate? 1) 14 7 = 2 2) You have a turtle on your back. 3) I like your flannel hat. 4) Mr. Saputo has 2300 records. 2

3 PROPOSIION NOAION We represent propositions by letters such as p, q, and r. or example, p: It always rains on uesdays. q: = 46 r: x is an even number. Negation Even though I have botched it a few times, negation means something different than A. his is the compliment. he negation of a proposition p is not p, and is written as p. he truth value of p is the opposite of the truth value of p. Example: he proposition p: It is raining outside. he negation of p, p: It is not raining outside. Write down an example about yourself! 3

4 Back to the example p: It is raining outside p: it is not raining outside. rom this example we can see that p is false when p is true true when p is false With this information, it is simple to organize the information in a ruth table. he first row is your propositions in general, followed by columns that that correspond with the truth value outcomes. Negation with sets or sets, the negation can also be the compliment. hey are one in the same, but the terminology is different. Example: U = 1 2, 3,5,7,9,11 ind the negation of x when x Z 4

5 NEGAION AND VENN DIAGRAMS We can use a Venn diagram to represent these propositions and their negations. or example, consider p: x is greater than 10. U is the universal set of all the values that the variable x may take. P is the truth set of the proposition p, or the set of values of x U for which p is true. P is the truth set of p. Example Consider U = x 2 < x < 12, x N and proposition p: x is a prime number. ind the truth sets of p and p, and display them on a Venn diagram. 5

6 COMPOUND PROPOSIIONS Compound propositions are statements which are formed using connectives such as and and or. Lets start with conjunctions: When two propositions are joined using the word and, the new proposition is the conjunction of the original propositions. If p and q are propositions, p q is used to denote their conjunction. Example p: Adriana had coffee for breakfast q: Adriana had bacon for breakfast p q: Adriana had coffee and bacon for breakfast. p q is only true if Adriana had both coffee and a bacon for breakfast, which means that both p and q are true. If either of p or q is not true, or both p and q are not true, then p q is not true. his leads to something very important: A conjunction is true only when both original propositions are true. 6

7 Lets use Adriana as an example We are going to look at a truth table to help us understand each situation that could possibly happen. Lets use Adriana as an example We can use Venn diagrams to represent conjunctions. Suppose: P is the truth set of p, and Q is the truth set of q. he truth set of p q is P Q, the region where both p and q are true. 7

8 Lets do another example. Write down p q for the following pairs of propositions and fill in a truth table: A. p: ranco is a student, q: ranco is a senior p q p q B. p: Heidi is a figure skater, q: Heidi is a profession sycronized swimmer. A. Which situation is true on the table? p q p q Lets do another example. A. Decide whether p q is true or false. Highlight the truth table that this circumstance follows. p: 4 is a factor of 16, q: 3 is a factor of 16. p q p q 8

9 DISJUNCION When two propositions are joined by the word or, the new proposition is the disjunction of the original propositions. If p and q are propositions, p q is used to denote their disjunction. Example: p: Daisy ran the 500m today q: Daisy ran the 1,000m today. p q: A disjunction is true when one or both propositions are true. A disjunction is only false if both propositions are false. he truth table for the disjunction p or q is: Here is the truth table for every disjunction p q p q 9

10 Venns If P and Q are the truth sets for propositions p and q respectively, then the truth set for p q is P Q, the region where p or q or both are true. he exclusive disjunction is true when only one of the propositions is true. he exclusive disjunction of p and q is written p q. We can describe p V q as p or q, but not both, or exactly one of p and q. Example: p: Orion ate cereal for breakfast q: Orion ate toast for breakfast p V q: Sally ate cereal or toast, but not both, for breakfast. he truth table for the exclusive disjunction p V q is: p q p q 10

11 he exclusive disjunction is true when only one of the propositions is true. he exclusive disjunction of p and q is written p q. We can describe p V q as p or q, but not both, or exactly one of p and q. Example: p: Orion ate cereal for breakfast q: Orion ate toast for breakfast p V q: Sally ate cereal or toast, but not both, for breakfast. If P and Q are the truth sets for propositions p and q respectively, then the truth set for p V q is the region shaded, where exactly one of p and q is true. p q P Q (P Q) p q p q Example 11

12 Do we have time for more? RUH ABLES AND LOGICAL EQUIVALENCE he truth tables for negation, conjunction, disjunction, and exclusive disjunction can be summarized in one table. p q p p q p q p q Construct a truth table for p q. We start by listing all possible combinations of p and q. We can fill in each column afterwards. p q q p q 12

13 AUOLOGY AND LOGICAL CONRADICION A compound proposition is a tautology if all the values in its truth table column are true. A compound proposition is a logical contradiction if all the values in its truth table column are false. 1) is p p a tautology a logical contradiction or neither? p p p p More difficult Show that ( q p) (q p) is a logical contradiction. 13

14 One last thing in C wo propositions are logically equivalent if they have the same truth table column. Lets do one alone Construct a truth table for the following propositions: 1) p q p q p p q 14

15 Lets break it down using the symbols = Both need to be true to have a true = One or more need to be true to have a true. = Only one can be true to have a true = he opposite of whatever it is in front of. hree propositions r is used to state a third proposition. At this point, what p, q and r are tend to get abstract, therefore, we are going focus on what happens in the truth table. p q r 15

16 You try to construct a 3 prop truth table for (p q) r Homework P.236 8B.1 #2, 3 P B.2 #2, 4, 5, 9, P.242 8C.1 #1-9 odd. P.244 8C.2 #2-6 even 16