GE 4 Group 8 Final Performance Task

GE 4 -B THE LANGUAGE GROUP 8 SILLIMAN UNIVERSITY CHAPTER II MINI-MAGAZINE ISSUE O1 MATHEMATICS IN THE MODERN WORLD OF

THE OF C an you imagine how you would be able to communicate with a seatmate in the bus who speaks an entirely different language from yours? Language facilitates communication and meaning-making. It allows people to express themselves and bridges the gap among people from various groups or cultural origins. Mathematics is also a language. It has its own symbol system, the same way the English or Greek languages have their own alphabet.

With this magazine, readers can dissect the essence of Mathematics in real life, particularly as an inclusive language that can be understood by all. Beyond the infinite numbers, complicated formulas, and the whole prejudice against this specific field, lies its usefulness in our day-to-day living. As researchers and writers, we took our alma mater’s Mathematics Club’s invitation to take part in their celebration of Mathematics Month as an opportunity to share interesting things with freshmen. Moreover, our team has come up with different exercises, trivia, puzzles, and such in order to showcase that one can have fun while learning Math. We hope to witness you enjoy as you flip each page after the other of this magazine! -Group 8

ttaabcleobolnfectoeontfnenttss introduction 05 the language of mathematics 05 learn these terms! 06 truth tables 08 conjunction and disjunction 08 conditionals, bi-conditionals & negation 09 exercises 10 find the word match logical statements 10 negation 11 construct and prove or disprove 12 propositions and connectives 13 transform conditional statements 14 symbolize sentences 15 16 comics answers 17 references 18 20

MA Every mathematical number or TH symbol has a corresponding word or phrase. Mathematics has a language of its own. – Roopa Banerjee -> LANGUAGE Mathematics is English everywhere, in every aspect of our lives. Years of itself. The studies and research have helped us understand the numbers universe in great detail. Globally, people are able to can represent grasp different concepts related to Mathematics nouns and the without translating them. Furthermore, numbers and operational signs can symbols are not the only things Mathematics is represent verbs. A associated with but intriguing words, as well. mathematical equation There is a word or phrase corresponding to every such as ‘2 x 3 = 6’ can be mathematical number or symbol. It can even be thought of as a sentence. viewed as simpler and more consistent than Just like the English language, Mathematics is based on grammar and correct syntax (Teaching Math as a Language, 2016). 05

BI-CONDITIONAL 06 Combination of a conditional DISJUNCTION statement and its converse written in the if and only if When a proposition is created form. Two line segments are by using the word \"or\" to join congruent if and only if they two other propositions. are of equal length. Disjuncts are the two statements in question. When CONDITIONAL at least one of its disjuncts is true, a disjunction is true; A propositional of the form: if otherwise, it is false. hypothesis, then conclusion. It is also known as an INVERSE implication or an if-then statement. Formed by negating both the hypothesis and conclusion: CONJUNCTION (If not p , then not q). When the word \"and\" is used MATHEMATICAL to combine two propositions EXPRESSIONS into one. Conjuncts are two propositions together. When When mathematical objects both of its conjuncts are true, like numbers, matrices, sets, a conjunction is true; vectors, functions, etc. are otherwise, it is false. properly joined together by well-defined operations, we CONTRAPOSITIVE call the resulting expression a mathematical expression. Formed by negating both the hypothesis and the MATHEMATICAL OBJECT conclusion and then (VARIABLE) switching places: (If not q, then not p). When a mathematical object's value or list of CONVERSE components are not specified, the mathematical Formed by simply switching object is typically referred to the hypothesis and the as a variable and can be conclusion: ( If q, then p). represented by a symbol, most frequently a letter from the English alphabet.

07 MATHEMATICAL SENTENCE RMS!EMATICS Also called a mathematical UNSOUND ARGUMENT statement, statement or proposal, is a sentence that An argument that is invalid can be identified as either or has at least one false true or false. premises. NEGATION THHE LAENGUSAGEEOF MTATHE The opposite of the given T mathematical statement. If “P” is a statement, then the negation of statement P is represented by ~P. The symbols used to represent the negation of a statement are “~” or “¬”. LEARN CHAPTER II: PROPOSITIONS SYMBOLIZING SENTENCES To be a sentence that is Logical statements are either true or false, but not conventionally denoted by both; sometimes called a lower case letters like p, q, statement or an assertion r, s, etc. SOUND ARGUMENT TRUTH TABLE An argument that is valid and has true premises. A diagram in rows and columns showing how the truth or falsity of a proposition varies with that of its components.

conjunctions p q p^q TTT TFF FTF FFF disjunctions p q pvq TTT TFT FTT FFF 08

conditionals →p q pq TTT TFF FTT FFT bi-conditionals ↔p q p q TTT TFF FTF FFT negation p ~p TF FT 09

DID YOU KNOW? Boolean algebra is the category of algebra in which the variable’s values are the truth values, true and false, ordinarily denoted 1 and 0 respectively. The important operations performed in Boolean ∧algebra are – conjunction ( ), ∨disjunction ( ) and negation (¬). Hint: Total of 15 words. We have finished discussing the important key terms that you may encounter for Chapter 2: The Language of Mathematics. To test your knowledge, find as many words as you can according to the terms defined in the previous pages. answers are found in page 18 10

MMAATTCCH MMAATTCCHH H Mtyaptcehoctfohlneodgfioitclilaoolnwsatilna, tgbeis-mctaoetnnedtmi(tceioonnntjasuli)nnicnCtioColonul,mudmnisnjAuBnw.cittihonit,s Column A Column B a, Conjunction 1.Today is Friday and tomorrow is b. Disjunction Saturday. 2.If two sides of a triangle are c. Conditional d. Bi-conditional congruent, then the angle opposite them are congruent. 3.I will pass the Math exam or I will be promoted . 4.If you will recite the poem ,then you will pass the oral examination. ⇔ DID YOU KNOW? The symbol “ ” works in mathematical logic and set theory the same way “=” works in arithmetic and algebra. In this case, the negative appears to distribute throughout the or statement by negating each statement individually and changes the or statement to an and statement. This is called De Morgan’s Law. answers are found in page 18 11

1.Select the statement that is the b. The speed limit is not 55 negation of “All summer days and granny is not driving 35. are muggy.” c. The speed limit is not a. All muggy days are summer. 55 or granny is driving 35. b. Some summer days are d. The speed limit is not 55 muggy. and granny is driving 35. c.Some summer days are not e. Counseling for road rage muggy. is available at 1 - 900 - d. No summer days are muggy. calmdown. 2.Select the statement that is the 5. Select the statement that is negation of “Some weasels are the negation of “All circus cuddly.” clowns are undignified.” a. No weasels are cuddly. a. All circus clowns are b. All weasels are cuddly. dignified. c.Some weasels are not b. All cirrus clouds are cuddly. indistinguishable. d.All cuttlefish are weasely. c.Some circus clowns are 3.Select the statement that is the dignified. negation of “Coach Spurrier is d.No circus clowns are charming and Coach Spurrier is dignified. modest.” 6. Select the statement that is a. Coach Spurrier is not the negation of “You wear charming and Coach matching socks to the interview Spurrier is not modest. or you don’t get hired.” b. Coach Spurrier is not a. You don’t wear matching charming or Coach Spurrier socks to the interview or is not modest. you get hired. c.Coach Spurrier is not b. You don’t wear matching charming and Coach socks to the interview and Spurrier is modest. you get hired. d.Let’s get serious for a c.You don’t wear matching minute. socks to the interview and 4. Select the statement that is the you don’t get hired. negation of “The speed limit is d.If you don’t wear matching 55 and granny is driving 35.” socks to the interview, then a. The speed limit is not 55 or you don’t get hired. granny is not driving 35. NEGA TION answers are found in page 18 12

CONSTRUTH TRUCT INMAKING TABLES TRUTH TABLES CONSTRUCT TRUTH TABLES FOR THE oAnclyonifjubnocPAttRihdOoinvsTajiuIslunPtceo:birtsfnuoiolayetfnnoh. tdrirpvsoaaupaornleuenlanyedilnyffsdiaqfaflfqsanoeldrsa. ere FOLLOWING STATEMENTS: ∧1. ~p ~q →2. ~q p → ∧ →3. (p q) (q p) PROVE OR DISPROVE THE FOLLOWING STATEMENT: → ∨The expressions P Q and ~P Q are logically equivalent. PRODVIESPORROVELOGICALEQUIVALENCY answers are found in page 18 13

Which of the following is/are not a proposition/s? 1.1 is a prime number. 2. Run. 3.Please stay at home. 4.π is a real number. DID George Boole (1815-1864) is considered the YOU “father of symbolic logic”. He developed logic as an KNOW? abstract mathematical system consisting of defined terms (propositions), operations (conjunction, disjunction, and negation), and rules for using the operations. Each of the following Represent the statements is a following statements proposition. Can you using logical tell which are true connectives. and which are false? 1.P or not Q. 2.If P and R, then Q. 1.5 <−8 3.P if and only if 2.The square of −1 (Q and R). is negative. 4.Not P and not Q. 3.2 +8=10 5.It is not the case 4.√2 is a rational that if P, then Q. number. 6.If P and Q, then R or S. answers are found in page 19 14

TRANSFORM THE FOLLOWING CONDITIONAL STATEMENTS INTO CONVERSE, INVERSE, AND CONTRAPOSITIVE STATEMENTS: If a triangle is acute, then all of its three angles are less than 90 degrees. The polygon is a triangle if the sum of the figure’s angles is equal to 180 degrees. The figure is a square only if the figure has four equal sides. The triangles have equal corresponding angles when the 15 triangles are congruent. The polygon is a square given that the polygon is quadrilateral. answers are found in page 19

YYYMMMBBBOOOLLLIIIZZZEEESSSYYYMMMBBBOOOLLLIIIZZZEEE Symbolize the paragraph: ItngicwitfnhnhiootevaevetvioloznelPl.eclirvvnrintnTeetevhhsmiddsozeeieldoevniignfgeennsondtotvotvhcceeifoeoinrrsorntrnrhrfcmaucfmueoiopcpgeueritctnoarniniouoolotttsunprndoyno.t, fitfpofTafratiirynhchrcreei.eeieaasarnirIlledlfsesifveotntanianhnhrroorteeegeet,t, living well. Using the paragraph above, write the following symbolic form into words: 1.(a^c) -> ~b 2. (~a) <=> (b^~c) answers are found in page 19 16

Math is the language of the Universe. So the more equations you know,the more you can converse with the cosmos. -Neil deGrasse Tyson 17

18 FIND THE WORD 3. PAGE 10 PROVE OR DISPROVE PAGE 13 PROPOSITIONS & CONNECTIVES PAGE 14 MATCH LOGICAL STATEMENTS PART I PAGE 11 Items 2 and 3 are not propositions. 1.A 3. B PART 2 2.D 4. C 1.False. A positive number is always greater than a negative NEGATION number. PAGE 12 2.False. The square of a negative number is always positive. 1.D 4. B 3. True. 2.C 5. A 4.False. √2 is not a rational number 3.A 6. A since it cannot be expressed as a quotient of two integers. CONSTRUCT TRUTH TABLES PAGE 13 PART III S) 1. ∨P ~Q ∧ →(P R) Q 2. ↔ ∧P (Q R) ∧~P ~Q →~(P Q) ∧ → ∨(P Q) (R

19 TRANSFORM CONDITIONAL Contrapositive: STATEMENTS PAGE 15 1.If not all of its three angles are less than 90 degrees, then a Converse: triangle is not acute. 1.If all of its three angles are less 2.If the polygon is not a triangle, than 90 degrees, then a triangle then the sum of the figure’s is acute. angles is not equal to 180 2.If the polygon is a triangle, then degrees. the sum of the figure’s angles is 3.If the figure does not have four equal to 180 degrees. equal sides, then the figure is not 3.If the figure has four equal sides, a square. then the figure is a square. 4.If the triangles are not congruent, 4.If the triangles are congruent, then the triangles do not have then the triangles have equal equal corresponding angles. corresponding angles. 5.If the polygon is not a 5.If the polygon is a quadrilateral, quadrilateral, then the polygon is then the polygon is a square. not a square. Inverse: SYMBOLIZE SENTENCES 1.If a triangle is not acute, then not PAGE 16 all of its three angles are less than 90 degrees. Part 1 2.If the sum of the figure’s angles is a: A President is a good president. not equal to 180 degrees, then b: The government officials are the polygon is not a triangle. involved in corruption. 3.If the figure is not a square, then c: The citizens of the country are the figure does not have four living well. equal sides. 4.If the triangles do not have equal ((a->~b)^(~b->c)^b)->~c corresponding angles, then the triangles are not congruent. 5.If the polygon is not a square, Part 2 then the polygon is not a 1.If a president is a good president quadrilateral. and the citizens of the country are living well, then the government officials are not involved in corruption. 2.A president is not a good president if and only if the government officials are involved in corruption and the citizens of the country are not living well.

20 REFERENCES Alt, M., Arizmendi, G., & Beal, C. (2014, July). The Relationship Between Mathematics and Language: Academic Implications for Children With Specific Language Impairment and English Language Learners. ASHA Wire. Retrieved July 25, 2022, from https://pubs.asha.org/doi/epdf/10.1044/2014_LSHSS-13-0003 Baltazar, Ragasa, & Evangelista, Mathematics in the Modern World, C&E Pub- lishing, 2018. Boolean Algebra. (2021, March 22). BYJUS. Retrieved July 24, 2022, from https://byjus.com/maths/boolean-algebra/ Garnier, Rowan, Discrete Mathematics: for New Technology, IOP Publishing Ltd 2002 George Boole. (n.d.). Encyclopedia Britannica. Retrieved July 24, 2022, from https://www.britannica.com/biography/George-Boole Ikenaga, B. (n.d.). Truth Tables, Tautologies, and Logical Equivalences. Bruce Ikenaga Mathematics. Retrieved July 24, 2022, from https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth- tables.html Libretexts. (2022, May 20). Basic Language of Mathematics (Exercises). Mathematics LibreTexts. Retrieved July 24, 2022, from https://math.libretexts.org/Courses/Mount_Royal_University/MATH_ 1150%3A_Mathematical_Reasoning/1%3A_Basic_Language_of_ Mathematics/1.E%3A_Basic_Language_of_Mathematics_(Exercises) Math Comics. (n.d.). Mathematrix. Retrieved July 24, 2022, from https://home.adelphi.edu/%7Estemkoski/mathematrix/comics.html Negative Statements. (2021, April 21). FlexBooks. Retrieved July 24, 2022, from https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts -2.0/section/16.3/primary/lesson/negative-statements-pcalc/

21 thteeam Group 8 GE 4-B DONNIELOU MAY THERESE VINCENT REGALADO REGIS BS Accountancy BS Accountancy

22 \"The strength of the team is each individual member. The strength of each member is the team.\" - Phil Jackson JERICHA SHEANDRA ANGELICA DALE LOU MAE RUIZ ROSALES SALATAN BS Pharmacy BS Accountancy BS Accountancy