GENERAL MATHEMATICS 11

Transcription

1 THE SEED MONTESSORI SCHOOL GENERAL MATHEMATICS 11 Operations on Propositions July 14, 2016

2 WORK PLAN Daily Routine Objectives Recall Lesson Proper Practice Exercises Exit Card

3 DAILY DOSE On ¼ sheet of yellow paper, answer item numbers 1-5 of Gear Up on page 198. Time Limit: 5 minutes

4 OBJECTIVES At the end of the period, you are expected to be able to: perform the different operations on propositions

5 STARTER Give one example of a compound proposition that does not make sense. Example: Let: r = it rains w = the soil gets wet d = President Duterte gets impeached S 1 : if r, then w S 2 : if r, then d S 1 makes sense while S 2 does not.

6 DEFINITION Logical operators words used to connect 2 or more propositions to form compound propositions

7 LESSON PROPER LOGICAL OPERATORS 1. Negation ( ~ ) 2. Disjunction ( v ) 3. Conjunction ( ^ ) 4. Implication ( )

8 NEGATION The negation of a proposition is its exact opposite. It is denoted by the symbol ~. Example: Let: P: 1 is an even number Its negation is represented by ~P (read as not P ) and is written as ~P: 1 is not an even number In simple English, we say that ~P: 1 is an odd number This is because of the dichotomy of evenness and oddness of numbers.

9 DEFINITION An event is said to be dichotomous if exactly one of two possible instances can happen at a time. DICHOTOMOUS EVENTS head or tail (in a single coin toss) on and off (lights) even or odd (numbers) NON-DICHOTOMOUS EVENTS prime or composite (numbers); 1 is neither prime nor composite citizenship walking and thinking equal or unequal

10 NEGATION Let P be a proposition. Other simple English translations of the negation of P are as follows: not P it is not the case that P it is false that P it is not true that P

11 EXAMPLES Negate each proposition. Express your answer in simple English. 1. A: I am under the STEM strand. ~A: I am not under the STEM strand. 2. B: The square of a negative number is positive. ~B: It is not the case that the square of a negative number is positive. 3. C: I need you more than I need oxygen. <3 ~C: It is false that I need you more than I need oxygen. </3

12 LESSON PROPER LOGICAL OPERATORS 1. Negation ( ~ ) 2. Disjunction ( v ) 3. Conjunction ( ^ ) 4. Implication ( )

13 DISJUNCTION The disjunction of two (or more) propositions is a compound proposition whose truth or falsity depends on the propositions themselves. Example: Let: P: 2 is a prime number Q: 2 is an even number Their disjunction is represented by P v Q (read as P or Q ) and is written as P v Q: 2 is a prime or an even number

14 DISJUNCTION Inclusive or: one or the other, or both I will eat hamburger or fries. Exclusive or: one or the other, but not both. A single toss of a coin will show a head or a tail. Unless otherwise stated, all disjunctions are inclusive by default.

15 DISJUNCTION Let P and Q be propositions. Some simple English translations of the disjunction of P and Q are as follows: P or Q either P or Q P unless Q

16 EXAMPLES Write the disjunction of the given propositions. Express your answer in simple English. 1. D: Yohan will take the Academic Track. E: Yohan will take the Tech-Voc Track. D v E: Yohann will take the Academic Track unless he takes the Tech-Voc Track. 2. F: Give me another chance. G: I will never love again. F v G: Give me another chance or I will never love again.

17 LESSON PROPER LOGICAL OPERATORS 1. Negation ( ~ ) 2. Disjunction ( v ) 3. Conjunction ( ^ ) 4. Implication ( )

18 CONJUNCTION The conjunction of two propositions implies the idea of both propositions taken at the same time. Example: Let: P: 2 is a prime number Q: 2 is an even number Their conjunction is represented by P ^ Q (read as P and Q ) and is written as P ^ Q: 2 is a prime and an even number In simple English, we say that 2 is an even prime number.

19 CONJUNCTION Let P and Q be propositions. Some simple English translations of the conjunction of P and Q are as follows: P and Q P moreover Q P although Q P still Q P furthermore Q P also Q P nevertheless Q P however Q P yet Q P but Q

20 EXAMPLES Write the conjunction of the given propositions. Express your answer in simple English. 1. H: The solutions of x 2 + 6x + 9 = 0 are equal. J: The solutions of x 2 + 6x + 9 = 0 are rational. H ^ J: The solutions of x 2 + 6x + 9 = 0 are equal and rational. 2. K: My brain says stop. L: My heart says go on. K ^ L: My brain says stop but my heart says go on.

21 LESSON PROPER LOGICAL OPERATORS 1. Negation ( ~ ) 2. Disjunction ( v ) 3. Conjunction ( ^ ) 4. Implication ( )

22 IMPLICATION An implication is also called a conditional statement or an if-then statement. The if part is called the antecedent (or hypothesis or premise) while the then part is called the consequent (or conclusion). Example: Let: S: I get a perfect score in the exam. T: I will treat you to lunch. The implication is represented by S T (read as if S, then T ) and is written as S T : If I get a perfect score in the exam, then I will treat you to lunch.

23 IMPLICATION Implications are directional. If the forward is true, the backward is not always true. Case 1: Backward is TRUE Example: P: The sum of two angles is 90. Q: Two angles are complementary. P Q: If the sum of two angles is 90, then they are complementary. The statement, If two angles are complementary, then their sum is 90. is also true.

24 IMPLICATION Implications are directional. If the forward is true, the backward is not always true. Case 2: Backward is FALSE Example: P: It rains. Q: The soil gets wet. P Q: If it rains, then the soil gets wet. However, the soil being wet does not necessarily mean that it rained. There might have been other factors that made the soil wet.

25 CONJUNCTION Let P and Q be propositions. Some simple English translations of the implication of P to Q are as follows: if P then Q P implies Q P is a sufficient condition for Q Q is a necessary condition for P Q if P Q follows from P Q provided P Q whenever P Q is a logical consequence of P

26 EXAMPLES Write an implication given the propositions. Express your answer in simple English. 1. M: All the angles of a triangle are acute. N: A triangle is acute. M N: If all the triangles of a triangle are acute, then the triangle is acute. 2. O: You think that I m still holding on. P: You should go and love yourself. O P: If you think that I m still holding on, then you should go and love yourself.

27 GROUPING MARKS Parentheses, brackets and braces are called grouping marks. The following are the guidelines in using them when logical operators are involved. 1. both and and either or Propositional Phrase Propositional Form both P or Q and R P or both Q and R either P and Q or R P and either Q or R (P v Q) ^ R P v (Q^R) (P^Q) v R P ^ (Q vr) * Insert grouping marks right after either or both.

28 GROUPING MARKS Parentheses, brackets and braces are called grouping marks. The following are the guidelines in using them when logical operators are involved. 2. neither-nor Propositional Phrase Propositional Form neither P nor Q not either P or Q ~ ( P v Q ) both P or Q are not

29 GROUPING MARKS Parentheses, brackets and braces are called grouping marks. The following are the guidelines in using them when logical operators are involved. 3. both not and not both Propositional Phrase Propositional Form P and Q are not both ~ ( P ^ Q ) P and Q are both not ~P ^ ~Q

30 SEATWORK Answer the following on ¼ sheet of yellow paper. Let: P: I am walking. Q: I am running. R: I am jogging. 1. ~P 2. ~P ^ Q 3. ~(P v Q) 4. ~R P 5. ~(Q v R)

31 ANSWERS Let: P: I am walking. Q: I am running. R: I am jogging. 1. ~P: I am not walking. 2. ~P ^ Q: I am not walking but running. 3. ~(Q v P): I am neither running nor walking. 4. ~R P: If I am not jogging, then I am walking. 5. ~(P ^Q): I am not both walking and running.

32 EXIT CARD Give one compound proposition that is related to your strand. STEM If a star explodes, then a black hole will be formed and it will consume everything that goes nearby. ABM If it is the 15 th or the 30 th of the month, then it is good to put some products on sale. HUMSS Both the Philippines and China may not prohibit each other in fishing in the West Philippine Sea if both countries are to abide by the decision of the International Tribunal.