GEED 10053 Mathematics in the Modern World Lecture Notes PDF

GEED 10053 MATHEMATICS IN THE MODERN WORLD Department of Mathematics and Statistics College of Science 1/70  Chapter 2 LANGUAGE OF MATHEMATICS OUTLINE 1. Propositions 2. Arguments 3. Sets 4. Quantifi...

GEED 10053 MATHEMATICS IN THE MODERN WORLD Department of Mathematics and Statistics College of Science 1/70  Chapter 2 LANGUAGE OF MATHEMATICS OUTLINE 1. Propositions 2. Arguments 3. Sets 4. Quantifications 2/70 LANGUAGE OF MATHEMATICS Learning Outcomes: At the end of the lesson, the students are able to: i. Determine and classify propositions; ii. Apply logical connectives in operating propositions; iii. Characterize sets and related notions iv. Perform set operations and solve related problems vi. Determine and classify quantifications of propositions over a set vi. Determine and stablish validity of arguments 3/70 D. QUANTIFICATION 4/70 Quantifications Propositional Function Let U be a set and x ∈ U. A propositional function or an open sentence p(x) is a declarative sentence that contains a variable x. We call U as the universe of discourse. Remark: Upon substitution of a constant c ∈ U, the propositional function becomes a proposition. 5/70 Quantifications Example 1 2. Let U = { vertebrates } and 1. Let U = Z and p(x) : x + 1 > 5. m(x) : x is a mammal. p(3) : m(spermwhale) : p(6) : m(vampirebat) : p(−10) : m(shark) : p(−1) : m(falcon) : 6/70 Quantifications 3. Let U = R and q(x) : x 2 ≥ 0. q(−π) : 5. Let U = { countries of the Earth } q(0) : and c(x, y ) : x and y come from the same continent.   1 q : 2 c(Thailand, Vietnam) : 4. Let U = R and r (x) : |x| < 0. c(Greece, Italy ) : √ c(Canada, Brazil) : r ( 2) : c(Egypt, Philippines) : r (0) : r (−3) : 7/70 Quantifications Quantification A quantification is a statement of extent to which a propositional function possesses a truth value over a subset of the universe of discourse U. Types of Quantification Let p(x) be a propositional function over a universe of discourse U. A universal quantification of p(x) is a quantification on all elements of U. An existential quantification of p(x) is a quantification on some elements of U. 8/70 Quantifications Comparison Between the Two Quantifications Word Quantifier Instance Instance Quantifiers Symbol of Truth of Falsity for all, Universal for every, ∀ p(x) is true p(x) is false Quantification for each, for all for at least one for any x ∈U x ∈U for some, Existential at least one, ∃ p(x) is true p(x) is false Quantification there exists for at least one for all x ∈U x ∈U 9/70 Quantifications Example 2 2. The statement 1. The statement All Asians are Filipinos. Every song is a poem. is a universal quantification. is a universal quantification. Let Let U = { Asians } U = {songs} f (x) :x is a Filipino. p(x) :x is a poem. In symbols, In symbols, ∀x ∈ U, f (x). ∀x ∈ U, p(x). This is a false universal This is a true universal quantification. quantification. 10/70 Quantifications 3. The statement 4. The statement The negative of any real number is a The square of every real number is real number. positive. is a universal quantification. is a universal quantification. Let Let U =R U =R n(x) : −x ∈ R. s(x) : x 2 > 0 Thus, Hence, ∀x ∈ R, −x ∈ R ∀x ∈ R, x 2 > 0 or ∀x ∈ U, n(x) or ∀x ∈ U, s(x) is a true universal quantification. is a false universal quantification. 11/70 Quantifications 6. The statement 5. The statement There is a month in a year that starts Some mammals are sea-dwellers. with Z. is an existential quantification. is an existential quantification. Let Let U = {mammals} U = { months in a year } s(x) :x is a sea-dweller z(x) :x is a word that starts with Z In symbols, In symbols, ∃x ∈ U, s(x). ∃x ∈ U, z(x). This is a true existential This is a false existential quantification. quantification. 12/70 Quantifications 7. The statement 8. The statement Some rational numbers There is a counting number between are less than 3. 3 and 4. is an existential quantification. is an existential quantification. Let Let U =Q U =N l(x) : x < 3. b(x) : 3 < x < 4 Thus, Hence, ∃x ∈ Q, x < 3 ∃x ∈ N, 3 < x < 4 or ∃x ∈ U, l(x) or ∃x ∈ U, b(x) is a true existential quantification. is a false existential quantification. 13/70 Quantifications Euler Diagrams An Euler diagram is the Venn diagram that is used to represent quantifications of arguments. 14/70 Quantifications Arguments over Quantification TYPE : Universal Affirmative FORMULATION : All p are q. TRUE EXAMPLE : All birds are vertebrates. FALSE EXAMPLE : Every integer is greater than 0. 15/70 Quantifications Arguments over Quantification TYPE : Universal Negative FORMULATION : No p are q. TRUE EXAMPLE : No real number has a negative square. FALSE EXAMPLE : There are no liquid metal elements. 16/70 Quantifications Arguments over Quantification TYPE : Existential Affirmative FORMULATION : Some p are q. TRUE EXAMPLE : Some Filipino writers have literary works in English. FALSE EXAMPLE : There is a counting number less than 0. 17/70 Quantifications Arguments over Quantification TYPE : Existential Negative FORMULATION : Some p are not q. TRUE EXAMPLE : At least one Senator of the Philippines is not a former House Representative. FALSE EXAMPLE : There are PUP students who graduated with two bachelor degrees in 2020. 18/70 Quantifications Evaluating Validity of Arguments using Euler Diagrams 1. Draw an Euler Diagram based on the premises. 2. The argument is invalid if there is a way to draw the diagram that makes the conclusion false. 3. The argument is valid if the diagram cannot be drawn to make the conclusion false. 4. If the premises are insufficient to determine the location of an element or a set mentioned in the conclusion, then the argument is invalid. 19/70 Quantifications Example 3 Determine the validity of the argument Every bird is a vertebrate. Parrots are birds. ∴ Parrots are vertebrates. 20/70 Quantifications Example 3 Determine the validity of the argument Every bird is a vertebrate. Parrots are birds. ∴ Parrots are vertebrates. 21/70 Quantifications Example 3 Determine the validity of the argument Every bird is a vertebrate. Parrots are birds. ∴ Parrots are vertebrates. 22/70 Quantifications Example 3 Determine the validity of the argument Every bird is a vertebrate. Parrots are birds. ∴ Parrots are vertebrates. 23/70  Quantifications Example 3 Determine the validity of the argument Every bird is a vertebrate. Parrots are birds. ∴ Parrots are vertebrates. ∴ The argument is valid. 24/70 Quantifications Example 4 Determine the validity of the argument Some juices have antioxidants. Citrus X is a juice. ∴ Citrus X has antioxidants. 25/70 Quantifications Example 4 Determine the validity of the argument Some juices have antioxidants. Citrus X is a juice. ∴ Citrus X has antioxidants. 26/70 Quantifications Example 4 Determine the validity of the argument Some juices have antioxidants. Citrus X is a juice. ∴ Citrus X has antioxidants. 27/70  Quantifications Example 4 Determine the validity of the argument Some juices have antioxidants. Citrus X is a juice. ∴ Citrus X has antioxidants. ∴ The argument is invalid. 28/70 Quantifications Example 5 Determine the validity of the argument Some singers are composers. No composer is a dancer. Melody is a singer-composer. ∴ Melody is not a dancer. 29/70 Quantifications Example 5 Determine the validity of the argument Some singers are composers. No composer is a dancer. Melody is a singer-composer. ∴ Melody is not a dancer. 30/70 Quantifications Example 5 Determine the validity of the argument Some singers are composers. No composer is a dancer. Melody is a singer-composer. ∴ Melody is not a dancer. 31/70 Quantifications Example 5 Determine the validity of the argument Some singers are composers. No composer is a dancer. Melody is a singer-composer. ∴ Melody is not a dancer. 32/70  Quantifications Example 5 Determine the validity of the argument Some singers are composers. No composer is a dancer. Melody is a singer-composer. ∴ Melody is not a dancer. ∴ The argument is valid. 33/70 Quantifications Example 6 Determine the validity of the argument All squares are rhombi. Every rhombus is a parallelogram. A parallelogram is a quadrilateral. ∴ All squares are quadrilaterals. 34/70 Quantifications Example 6 Determine the validity of the argument All squares are rhombi. Every rhombus is a parallelogram. A parallelogram is a quadrilateral. ∴ All squares are quadrilaterals. 35/70 Quantifications Example 6 Determine the validity of the argument All squares are rhombi. Every rhombus is a parallelogram. A parallelogram is a quadrilateral. ∴ All squares are quadrilaterals. 36/70 Quantifications Example 6 Determine the validity of the argument All squares are rhombi. Every rhombus is a parallelogram. A parallelogram is a quadrilateral. ∴ All squares are quadrilaterals. 37/70 Quantifications Example 6 Determine the validity of the argument All squares are rhombi. Every rhombus is a parallelogram. A parallelogram is a quadrilateral. ∴ All squares are quadrilaterals. 38/70  Quantifications Example 6 Determine the validity of the argument All squares are rhombi. Every rhombus is a parallelogram. A parallelogram is a quadrilateral. ∴ All squares are quadrilaterals. ∴ The argument is valid. 39/70 Quantifications Example 7 Determine the validity of the argument Some dietitians are overweight. No gym trainers are overweight. ∴ No gym trainers are dietitians. 40/70 Quantifications Example 7 Determine the validity of the argument Some dietitians are overweight. No gym trainers are overweight. ∴ No gym trainers are dietitians. 41/70 Quantifications Example 7 Determine the validity of the argument Some dietitians are overweight. No gym trainers are overweight. ∴ No gym trainers are dietitians. 42/70 Quantifications Example 7 Determine the validity of the argument Some dietitians are overweight. No gym trainers are overweight. ∴ No gym trainers are dietitians. 43/70  Quantifications Example 7 Determine the validity of the argument Some dietitians are overweight. No gym trainers are overweight. ∴ No gym trainers are dietitians. ∴ The argument is invalid. 44/70 Quantifications Example 8 Determine the validity of the argument Every tabloid is a newspaper. No magazine is a newspaper. ∴ There are some magazines that are tabloids. 45/70 Quantifications Example 8 Determine the validity of the argument Every tabloid is a newspaper. No magazine is a newspaper. ∴ There are some magazines that are tabloids. 46/70 Quantifications Example 8 Determine the validity of the argument Every tabloid is a newspaper. No magazine is a newspaper. ∴ There are some magazines that are tabloids. 47/70 Quantifications Example 8 Determine the validity of the argument Every tabloid is a newspaper. No magazine is a newspaper. ∴ There are some magazines that are tabloids. 48/70  Quantifications Example 8 Determine the validity of the argument Every tabloid is a newspaper. No magazine is a newspaper. ∴ There are some magazines that are tabloids. ∴ The argument is invalid. 49/70 Quantifications Negation of a Quantification 1. Negation of a Universal Quantification ∼ (∀x ∈ U, p(x)) ⇔ (∃x ∈ U, ∼ p(x)) Not all x ∈ U satisfies p(x) ⇔ Some x ∈ U do not satisfy p(x) 2. Negation of an Existential Quantification ∼ (∃ ∈ U, p(x)) ⇔ (∀x ∈ U, ∼ p(x)) No x ∈ U satisfies p(x) ⇔ Every x ∈ U do not satisfy p(x) 50/70 Quantifications Example 9 1. The quantification Every real number has a square greater than or equal to 0 can be expressed in symbols as ∀x ∈ R, x 2 ≥ 0. Its negation when written in symbols is ∃x ∈ R, x 2 < 0. Thus, its negated form is Some real numbers have squares less than 0 or Some real numbers have negative squares. 51/70 Quantifications 2. The quantification All sea dwellers are fish. Its negation is Some sea dwellers are not fish. 52/70 Quantifications 3. The quantification At least one plant does not bear a flower. Its negation is Each plant bears a flower. 53/70 Quantifications 4. The quantification There exists a rock that can fly high. Its negation is Every rock can’t fly high or No rock can fly high. 54/70 Quantifications 5. The quantification No superhero is a billionaire is the same with Every superhero is not a billionaire. Its negation is Some superheroes are billionaires. 55/70 Quantifications Evaluating Validity of Arguments using Euler Diagrams 1. Draw an Euler Diagram based on the premises. 2. The argument is invalid if there is a way to draw the diagram that makes the conclusion false. 3. The argument is valid if the diagram cannot be drawn to make the conclusion false. 4. If the premises are insufficient to determine the location of an element or a set mentioned in the conclusion, then the argument is invalid. 56/70 Quantifications Evaluating Validity of Arguments using Euler Diagrams 1. Draw an Euler Diagram based on the premises. 2. Draw an Euler Diagram based on the negation of the conclusion. 3. If the Euler Diagram of the negated conclusion fits in the Euler Diagram of the premises, then it is invalid. 4. If the Euler Diagram of the negated conclusion does not fit in the Euler Diagram of the premises, then it is valid. 57/70 Quantifications Example 10 Determine the validity of the argument The negation of the conclusion is Every bird is a vertebrate. Parrots are not vertebrates. Parrots are birds. ∴ Parrots are vertebrates. 58/70 Quantifications ∴ The argument is valid. 59/70 Quantifications Example 11 Determine the validity of the argument The negation of the conclusion is Some juices have antioxidants. Citrus X is a juice. Citrus X has no antioxidants. ∴ Citrus X has antioxidants. 60/70 Quantifications ∴ The argument is invalid. 61/70 Quantifications Example 12 Determine the validity of the argument Some singers are composers. The negation of the conclusion is No composer is a dancer. Melody is a dancer. Melody is a singer-composer. ∴ Melody is not a dancer. 62/70 Quantifications ∴ The argument is valid. 63/70 Quantifications Example 13 Determine the validity of the argument All squares are rhombi. The negation of the conclusion is Every rhombus is a parallelogram. Some squares are not quadrilaterals. A parallelogram is a quadrilateral. ∴ All squares are quadrilaterals. 64/70 Quantifications ∴ The argument is valid. 65/70 Quantifications Example 14 Determine the validity of the argument The conclusion is the same with Some dietitians are overweight. No gym trainers are overweight. Every gym trainer is not a dietitian. ∴ No gym trainers are dietitians. Its negation is Some gym trainers are dietitians. 66/70 Quantifications ∴ The argument is invalid. 67/70 Quantifications Example 15 Determine the validity of the argument The negation of the conclusion is Every tabloid is a newspaper. No magazine is a newspaper. Every magazine is not a tabloid. ∴ There are some magazines that It is the same with are tabloids. No magazine is a tabloid. 68/70 Quantifications ∴ The argument is invalid. 69/70 References 1. Angel, A.R., Abbott, C.D. & Runde, D.C. (2012). A Survey of Mathematics with Applications. Pearson Education Inc. 2. Lippman, D. (2017). Math in Society. Pierce College Ft Steilacoom 3. Rosen, K.. (2012). Discrete Mathematics and Its Applications. McGraw Hill 4. Smith, K.J. (1998). Nature of Mathematics. Brooks/Cole Publishing Company 5. Statzkow, R. & Bradshaw, R. (1995). The Mathematical Palette. Saunders College Publishing 70/70