Genmath 11 logic PDF

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General MathematicsModule 10LOGICLogic has been studied since the classical Greek period (600 – 300 BC). The Greeks, most notably Thales, were the first to formally analyze the reasoning process. Aristotle (384 – 322BC), the “father of logic”, and many other Greeks searched for universal truths that...

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General Mathematics Module 10

LOGIC Logic has been studied since the classical Greek period (600 – 300 BC). The Greeks, most notably Thales, were the first to formally analyze the reasoning process. Aristotle (384 – 322BC), the “father of logic”, and many other Greeks searched for universal truths that were irrefutable. A second great period for logic came with the use of symbols to simplify complicated logical arguments. George Boole (1815 – 1864) is considered the “father of symbolic logic”. He developed logic as an abstract mathematical system consisting of defined terms (propositions), operations (conjunction, disjunction, and negation), and rules for using the operations. This module discusses the key concepts of propositional logic, syllogisms, and fallacies in real – life arguments.

What I Know (Prete (Pretest) st) Direction: Choose the letter of the best answer and write this on your answer sheet. 1. Which of the following sentences is a proposition? a. The man is faithful and responsible. b. Am I ready? c. STOP! You are not allowed to leave the house. d. Buy one kilo of rice when you go to the market. 2. Conjunction is a compound proposition connected by the word ______? a. and c. if … then b. or d. if and only if 3. Disjunction proposition can be written in symbol as a. ~p c. p ˄ q b. p → q d. p ˅ q 4. A statement: “It is not the case that Joemar can play the piano or Ariel can play the guitar” can be written in symbolic statement as a. p ˅ q c. ~ (p ˅ q) b. p ˄ ~ q d. ~ (p ˄ q) 5. If constructing the truth table of the statement (p → q) ↔ (~p → q), which will be the result in the last column? a. T, T, T, T c. T, T, F, F b. T, F, T, F d. F, F, F, F 6. The truth table final answer if p is true and q is false for inclusive disjunction is? a. True c. maybe b. False d. cannot be determined

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7. Under what condition or conditions is a conjunction true? a. When both p and q are true. b. When p is true, and q is false. c. When p is false, and q is true. d. When both p and q are false. 8. Let p be the statement: “Jing is a senior high school student” and let q be “Jing is over 15 years old”. Under which condition or conditions is the statement: “Jing is neither a senior high school student nor over 15 years old” true a. ~ p ˄ q is true when at least one of p and q is true. b. ~ p ˄ ~ q is true when both p and q are false. c. ~ p ˄ q is true when at least one of p and q are false. d. ~ p ˄ ~ q is true when both p and q are true. 9. Write in words the converse of the statement: “If n + 1 is an odd number, then 3 is a prime number” a. “If n + 1 is not an odd number, then 3 is not a prime number” b. “If 3 is not a prime number, then n + 1 is not an odd number” c. “If 3 is a prime number, then n + 1 is an odd number” d. “It is the case that n + 1 is not an odd number, then 3 is not a prime number” 10. The proposition p and q that have the same truth values are said to be a. Converse of each other c. Inverse of each other b. Contrapositive to each other d. Equivalent to each other 11. Which of the following is not a tautology? a. Modus Ponens c. Modus Tollens b. Affirming the Disjunct d. Double Negation 12. Translate the argument into symbolic form. If Joanna is relaxed, she is productive. If she is productive, she is happy. Therefore, she is not happy, then she is not relaxed. a. p → q b. ~ p ˅ q c. p ˅ ~ q d. p ↔ q ~p p ~q q↔r  q  q  ~p  ~r → ~ p 13. An argument is valid if it satisfies the validity condition that: a. The premises p1, p2, …, pn are false, and the conclusion is false. b. The premises p1, p2, …, pn are either true or false, and the conclusion is true. c. The premises p1, p2, …, pn are either false or true, and the conclusion is false. d. The premises p1, p2, …, pn are true, and the conclusion is true. 14. The following are fallacies in logic except a. Law of Syllogism c. Improper Transposition b. Affirming the Disjunct d. Denying a Conjunct 15. The argument “if it rains today, then 3 + 3 = 6. It rained today. Therefore, 3 + 3 = 6 is valid because a. It is a fallacy of the converse c. By rule of addition b. It is a Modus Ponens d. By rule of proof by cases

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Lesson

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Pr Propositions opositions and Symbols

What I Need to Know After going through this module, you are expected to: 1. Illustrates and symbolizes propositions. 2. Distinguishes between simple and compound propositions. 3. Performs the different types of operations on propositions.

What What’’s In Identify and label each sentence as declarative, interrogative, imperative, or exclamatory. 1. 2. 3. 4. 5.

Frontlines are considered heroes during COVID–19 pandemic. Stop COVID-19! Why does it spread that fast? Have you heard of the good news? Please stay home.

Sentences classified according to the purpose of the speaker or writer: 1. Declarative Sentence makes a statement and give information that normally end with a full – stop/period. 2. Interrogative Sentence ask a question of something or some information and always end with a question mark. 3. Imperative Sentence give a command telling someone to do something and end with a full-stop/period or exclamation mark/point. 4. Exclamative Sentence express strong emotion/surprise – an exclamation – and always end with an exclamation mark/point.

What What’’s New Activity 1 Determine whether this sentence answerable by True or False. 1. Department of Education announces this school year class start on August 24. 2. May 30, 2021 will be the last day of school. 3. When will you allowed to go to school? 4. Answer pages 5 to 15 of this module. 5. log 2 2 = 1

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Definition 1: A proposition is a declarative sentence that is either true or false, but not both. Notation: Variables are used to represent propositions. The most common variables used are p, q, and r. If a proposition is true, then its truth value is true, which is denoted by T; otherwise, its true value is false, which is denoted by F.

What is It In the Activity, sentences 1, 2 and 5 are proportions. The truth or falsity of each can be determined by a direct check. However, sentence 3 and 4 cannot be answered as being true or false. Sentence 3 is a question (interrogative sentence) and sentence 4 is a command (imperative sentence). There are some sentences that are not propositions. 1. “Do you want to go to the movies?” Since a question is not a declarative sentence, it fails to be a proposition. 2. “Clean up your room.” Likewise, an imperative is not a declarative sentence; hence, fails to be a proposition. 3. “2x = 2 + x.” This is a declarative sentence, but unless x is assigned a value or is otherwise prescribed, the sentence neither true nor false, hence, not a proposition. 4. “this sentence is false.” What happens if you assume this statement is true? False? This example is called a paradox and is not a proposition, because it is neither true nor false.

Wh What at at’’s More Enrichment Activities Activity 2

Decide whether each of the following is a proposition or is not a proposition.

1. Rudrego R. Dutente is the president of the Philippines. 2. 6 x 6 = 36 3. The number 4 is even and less than 12 4. Ouch! 5. What time is it? 6. x + 2 = 4 7. y – z = z – y 8. Joan’s solution is incorrect. 9. Rheza Mae is passing Statistics 10. Rheza Mae is passing Statistics but she is falling in 21st Century Literature.

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Solution: Question no. 1, 2 and 3 are propositions, because each of them is either true or false (but not both). Question no. 4 and 5 are not propositions because they do not proclaim anything; they are exclamation and question, respectively. Question no. 6 is not a statement because we cannot tell whether it is true or false unless we know the value of x. It is true when x = 2; it is false for other x – values. Since the sentence is sometimes true and sometimes false, it cannot be a statement. Question no. 7 since y – z = z – y, is sometimes true and sometimes false; it cannot be a statement. Question no. 8 is not a proposition. It is a paradox. It is neither true nor false. Question no. 9 is a proposition and an example of a simple proposition. Question no. 10 is a compound proposition. It is a combination of two simple propositions “Rheza Mae is passing Statistics” and “she is falling in 21 st Century Literature”. Definition 2: Simple Proposition – a proposition that conveys one thought with no connecting words. Example: “2 is an even number” “A square has all its sides equal” Compound Proposition – contains two or more simple propositions that are put together using connective words. Example: “11 is both an odd and prime number” can be broken into two propositions. “11 is an odd number” and “11 is a prime number” so it is a compound statement. Simple proposition can be combined to form compound propositions by using logical connectives or simply, connectives. Words such as and, or, nor and if… then are example of connectives. Basic Logical Connectives If the proposition is compound, then it must be one of the following: conjunction, disjunction, conditional, biconditional, or negation. Conjunction If two simple propositions p and q are connected by the word ‘and’, then the resulting compound proposition “p and q” is called a conjunction of p and q and is written in symbolic form as “p ˄ q”. Example: Form the conjunction of the following simple propositions: p: Jethro is a boy. q: He is a grade 5 pupil. Solution:

The conjunction of the proposition p and q is given by p ˄ q: Jethro is a boy and he is a grade 5 pupil.

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Disjunction If two simple propositions p and q are connected by the word ‘or’, then the resulting compound proposition “p or q” is called a disjunction of p and q and is written in symbolic form as “p ˅ q”. Example: p: q:

Form the disjunction of the following simple propositions: Joyce will pass all her subject. She will be retained

Solution:

The disjunction of the proposition p and q is given by p ˅ q: Joyce will pass all her subject or she will be retained.

Conditional If two simpl e propositions p and q are joined by a connectivity ‘if then’, then the resulting compound proposition “if p then q” is called a conditional proposition or an implication and is written in symbolic form as “p → q” or “p  q”. Here, p is called hypothesis (or antecedent) and q is called conclusion (or consequent) of the conditional proposition (p  q). The conditional p →q may also be read “p implies q”. Example: 1. 2. 3. 4.

If Ariel work hard then he will be successful. If you eat more then you lost your diet. If ABC is a triangle, then A + B + C = 180ᵒ. If Jonathan is in Grade 11, then he is a Senior High School student

Biconditional If two proposition p and q are connected by the connective ‘if and only if” then the resulting compound proposition “p if and only if q” is called a biconditional of p and q and is written in symbolic form as p ↔ q. The proposition may also be written as “p iff q”. The propositions p and q are the components of the biconditional. Example: 1. “Two sides of a triangle are congruent if and only if two angles opposite them are congruent. 2. Larseny is a STEM student if and only if she likes Science. Negation An assertion that a statement fails, or denial of a statement is called the negation of the statement. The negative of a statement is generally formed by introducing the word “not” at some proper place in the statement or by prefixing the statement with “it is not the case that” or “It is false that”. The negation of a statement p in symbolic form is written as “~p”. Example: Write the negation of the statement. p: School year 2020 – 2021 will start on June 1.

or or

Solution: The negation of p is given by ~p: S chool year 2020 – 2021 will not start on June 1. ~p: It is not the case that the school year 2020 – 2021 will start on June 1. ~p: It is false that the school year 2020 – 2021 will start on June 1.

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Negation of the Disjunction p or q ~ (p ˅ q) means ~ p ˄ ~ q Negation of the Disjunction p or q ~ (p ˄ q) means ~ p ˅ ~ q Example: Let p represent the proposition “Bang is beautiful”, and Let q represent the proposition “Joan is cute”. 1. Write the following in symbols and then in words. a. The conjunction of the negation of p and q Solution: In symbol: The negation of p and q are ~ p and ~ q, respectively. Thus, the conjunction of the negations of p and q in symbols is ~ p ˄ ~ q. In words: Bang is not beautiful, and Joan is not cute. b. The disjunction of the negation of p and q Solution: In symbol: ~ p ˅ ~ q. In words: Either Bang is not beautiful, or Joan is not cute. c. The negation of the conjunction of p and q Solution: In symbol: ~ (p ˄ q). In words: It is not the case that Bang is beautiful and Joan is cute. d. The negation of the disjunction of p and q Solution: In symbol: ~ (p ˅ q). In words: It is not the case that either Bang is beautiful, or Joan is cute. Symbols If x and y are the frequently used letters in algebra, the letters p, q, or r are often used to represent propositions in logic. The table below shows the several symbols for connectives, together with the respective types of compound proposition. Table 1 : Types of compound proposition Connective Symbol and ˄ or ˅ not ~ If … then → If and only if ↔ (iff)

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Type of Statement Conjunction Disjunction Negation Conditional Biconditional

Convert each compound proposition into symbols.

Activity 3

Let p represent the proposition “Facebook is a source of information.” q represent the proposition “Social Media spread fake news.” a. b. c. d. e.

Facebook is a source of information and social media spread fake news. Facebook is not a source of information or social media do not spread fake news. It is not the case that social media spread fake news and Facebook is a source of information. If Facebook is a source of information, then social media spread fake news. Social Media spread fake news if and only if Facebook is a source of information.

Solution: a. d.

p˄q p→q

Activity 4

a. p ˅ q

b. ~p ˅ ~q e. q ↔ p

c. ~q ˄ p

Let p represent the proposition “Television network remains shut down” and q represent the proposition “The network paid their taxes”. Write each symbolic statement in words. b. ~q ˄ p

c. p→q

d. ~ (p ˄ q)

e. ~ (q ˅ p)

Solution: a. b. c. d.

Television network remain shut down or the network paid their taxes. The network did not pay their taxes and television network remain shut down. If television network remains shut down, then the network paid their taxes. It is not the case that television network remains shut down and the network paid their taxes. Another solution of letter d: by Negation of the Conjunction p and q Television network did not remains shutting down or the network did not pay their taxes. e. It is not the case that the network paid their taxes or television network remains shut down. Another solution of letter e: by Negation of the Disjunction p or q The network did not pay their taxes and the television network did not remains shutting down. Note: Commas indicates which simple statement are grouped together. Parentheses in symbolic statements are used to tell what type of statements are being considered. If there are no parentheses, we follow the dominance of connectives (Biconditional, Conditional, Disjunction or Conjunction (equal in value), and last will be Negation. Example: a. Irene is a Math teacher (d) or Science teacher (e), and work at Talisayan National High School (f). Solution: (d ˅ e) ˄ f

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b. Identify the symbolic statement as conjunction, disjunction, negation, conditional or biconditional. Solution: W ˄ (S → T) Conjunction: The parentheses separate the statement “˄” connective. W˅S↔T Biconditional: The double arrow is the dominant connective and there are no parentheses.

What I Have learned Classify each proposition as simple or compound. Classify each compound proposition as conjunction, disjunction, conditional, or biconditional, and negation. My daughter will take a nursing course. His uncle’s name is not Manny. A positive integers n is divisible by 3, if and only if, the sum of the digits of n is divisible by 3. Either Hazel want to go to Davao or she wants to visit her mother in Misamis Oriental. Mich is diligent and intelligent student.

Activity 5

1. 2. 3. 4. 5.

Wh What at I Can Do Application: Draw a Venn Diagram Showing the relationship of the following proposition. Write the each in statement form (1 – 3) and the symbolic form (4 – 5), draw and shade the corresponding area in the Venn diagram representing the operation. Use different colors in each question. Let A be the proposition representing “student like HUMSS strand” Let B be the proposition representing “student taking SMAW strand” Let C be the proposition representing “student pursuing STEM strand” 1. 2. 3. 4. 5.

A˄B C˅A ~C It is not the case that the student like HUMSS strand or student taking SMAW strand. Student like HUMSS strand and taking SMAW, or pursuing STEM strand.

Additi Additiona ona onall Ac Activities tivities Let M = Practical Research is difficult, A = P.E. is easy, T = Earth and Life is interesting and H = Oral Communication needs confidence. Write each statement in words. a. M ˄ (A ˅ H) b. T ˅ (M ˄ A) c. H ˄ A → T d. ~ (M ˄ H) e. A ˄ (T ˅ ~ H)

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Truth Tables

Lesson

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What hat I Need to Know After going through this module, you are expected to: 1. Determine the truth values of proportions.

What What’’s In Let O = Kenny is loyal and N = Jefred is loyal. Write each statement in symbolic form. a. b. c. d. e.

If Jefred is Loyal, then Kenny is not loyal Both Kenny and Jefred are loyal. Either Kenny or Jefred is loyal. Jefred is not loyal or Kenny is not loyal. It is not the case that Jefred is loyal or Kenny is loyal.

Solution: Let O = Kenny is loyal a. N → O b. O ˄ N c. O ˅ N d. ~N ˅ ~O e. ~ (N ˅ O)

Let N = Jefred is loyal

Recall: Table 2: Types of Statements and their Connective Symbol Connective And Or Or Not If … then If and only if (iff)

Symbol ˄ ˅

Type of Statement Conjunction Disjunction Exclusive Or Negation Conditional Biconditional

~ → ↔

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Meaning “p and q” “p or q (or both)” p q “not p” “if p then q” “p if and only if q”

Connectives in their dominant order: 1. Biconditional (↔) 2. Conditional (→) 3. Conjunction (˄) and Disjunction (˅) 4. Negation (~)

What What’’s New Activity 1 1. 2. 3. 4. 5.

Answer the following algebra question by True or False

((22)3) = 25 The additive inverse of – 10 is 10. The absolute value of a real number is negative. 5 > 10 or 5 < 7 320 + 320 + 320 = 321

What is It In the Activity 1, with the following algebra question no. 2,4 and 6 are true. While, question no. 1 (26) and 3 (positive or zero) are false. In Algebra, unknown values are represented by letters of the alphabet which we call variables while fixed numbers are called constant. In logic, variables are also use to present proportions in the same way that we use variables to represent numbers in algebra. A value in logic has only TRUE or FALSE for its value. In fact, true and false are the “numerical constants” in logic.

Definition 1: The truth table displays the relationship between the possible truth values of the propositions. Steps in Constructing Truth Tables: a. Recall dominant connectives and the use of parentheses then determine if the result is a conjunction, disjunction, or negation. b. Complete the columns under 1. The simple statements (P, Q, …) 2. The connectives negations inside parentheses 3. Any remaining statements and their negations 4. Any r...