Mathematical Language PDF

MMW REVIEWER: Mathematical Language Characteristics of Mathematical Language The difference between language of everyday life to language in mathematics Comparison: Mathematical language has expressions and sentences. We express problems in words, not in symbols In math,...

MMW REVIEWER: Mathematical Language Characteristics of Mathematical Language The difference between language of everyday life to language in mathematics Comparison: Mathematical language has expressions and sentences. We express problems in words, not in symbols In math, we need to translate words into symbols, before we can solve them. An Expression in Math A Sentence in Math 5+1 5+1=6 Reason: Reason: Only has operations (+, -, x, /, Has: and etc) = >, < ≥, ≤  Mathematical expression consists of terms. Examples: The term of a mathematical expression contains a number, and a letter separated by at least one of the fundamental operations. It does NOT state a complete thought. Thus, it does not make sense to ask if an expression is true or false. Mathematical expression is the counterpart of noun in English. Examples of nouns: persons, places, things. In algebra, variables or letters are used to represent unknown quantities. So x should be the variable. It’s also called literal coefficient Meanwhile, 5 is the coefficient. Also known as the numerical coefficient Meanwhile, 5 in the same expression is called constant whose value is irreplaceable. Conventions in Math Another form of a mathematical symbol used when quantities take different values is variables. A variable is a symbol commonly represented by any letter that may assume various values. For instance, the phrase “a number” is sometimes expressed as variable x, a, b, or any other letter in the English alphabet 1. x+10 2. 6 + X 3. xy (reason: If variables are in 1 term, it’s assumed it’s multiplication automatically) 4. ½ * (x+y) 5. ½ * (x+y) 6. 5-x 7. x = Helpful Notes for Math as a Language: (Na tamad ako)  SETS Mga bros: Georg Ferdinand Wala lang sa PPT (Search ka sa google, hahaha) Ludwig Philipp Cantor Sets Builder notation Roaster/Tabular Method Rule/Descriptive Method  False (8 is not prime) True False (2 is a prime) KINDS OF SET RELATIONSHIP OF SETS PRACTICE: |A| means cardinality; therefore, we count the amount of elements. It’s “7” “3” Reason: Because 0, 1 is encased in a (). So you count them as 1 No, they are equal in Cardinality, but they aren’t equal as in the elements themselves No (Just no) Venn Diagram Also called: Primary Diagram Set Diagram Logic Diagram Is a diagram that shows all possible logical relations between a finite collection of different sets. Union Intersection Difference Complement or and excluding (Wala talaga na word) – Sir Including overlapping But not Either both without Together with Shared by Remainder of Combined with Left after removing Not in  T u V = {1,2,3,4,5,7,9} V u W = {1,2,3,4,5,7,9} C ∩ D = {6, 9} D ∩ E = {2, 3, 9} F – G = {2, 4, 9} G – F = {1, 3, 7} A’ = (2,3,4,6,8} Example:  1. Step-by-Step with Explanation: 1. Draw a Venn Diagram and name each circle: 2. Answer the middle first. (Good thumb to always answer from middle to outer). Reason: Because we instantly know from the given that out of the 1000 engineering students. 272 passed. We can write the set as: Because only 272 students intersect the Physics, Chemistry, Mathematics. Extra reasoning: Key word: “And” 3. Answer the surroundings in the middle (Answering between P and C but not M, or (P ∩ C) - M) Reason: From the given, It says: 416 Students who’ve passed physics and chemistry. 272 students who’ve passed all 3 subjects (M ∩ P ∩ C) Those 272 have also passed physics and chemistry (M ∩ P ∩ C). Yes? Here, it will look like this: Students who’ve passed physics and chemistry: 416 Students who’ve passed all 3 subjects (M ∩ P ∩ C): 272  Students who’ve passed physics and chemistry but not math ((P ∩ C) - M): (?) So we have the 1st image and 2nd image; therefore, we should just minus them to get (P ∩ C) – M 4. You solve the surroundings the same way 5. Notes: You start with the middle part, which is those who passed Math, Physics, Chemistry (M ∩ P ∩ C. LOGIC Negation Answer: 1. Manila is not the capital of the Philippines 2. False Notes for Negation: 1. If the preposition starts with (p) “Manila is not the capital of the Philippines.” It’s negation (-p or ¬p) is: “Manila is the capital of the Philippines. Basically, take the opposite of the proposition 2. Its symbols can be – and ¬ Disjunction Answer: 1. Manila is the capital of the Philippines or 2+1 = 5 2. True Notes for Disjunction: 1. At least one of the propositions is TRUE 2. Keyword: or Conjunction Answer: 1. Manila is the capital of the Philippines and 2+1=5. 2. False Notes for Conjunction: 1. Both propositions must be TRUE 2. Keyword: and, but, while, yet, still Implication or Conditional Answer: 1. If Manila is the capital of the Philippines, then 2+1=5 2. False Structure: p = premise (the one being pointing) q = conclusion (the one being pointed at) Notes for Conjunction 1. How to know if it’s true a. If the conclusion is true, then the conditional statement is true automatically it is true  b. If both p and q are the same value (if both are either true or false) then the conditional statement is true 2. The keywords:  If it starts with P If it starts -> with q Translate this to standard conditional statement (If p, then q) If John takes calculus, then he has a sophomore, junior. or senior standing If she studies hard, then Mary will be a good student. If you sing, then my ears will hurt. If Ginebra were to win the championship, then LA Tenorio has to scare high. Converse, Contrapositive, Inverse If q -> p, then p ->q If p-> q, then q -> p (Just switch) You will do converse, and then do negation (Switch, Negate) You will just negate (Negate) Biconditional Answer: 1. Manila is the capital of the Philippines if and only if 2+1=5 2. False Notes: 1. How to know if it is true. a. As long as they are the same value (If both are true or both are false) then condition is true Additional Challenge: 1. False 2. True a. (Reason: As long as the conclusion is true on the condition, then it is true) 3. True a. (Reason: Same reasoning with 2.) 4. True a. (Because the premise is false, the conclusion is either true or false. i. If it is F -> F, Then it is true (Reason: If both premise and conclusion is true or false, then condition is true) ii. If it is F -> T, then is is true (Reason: same with 2.) Truth Table: Example: Notes: For simple propositions. Make sure you give every instance of T and F. Reasoning: How the table is structured: 1. Left to right 2. You start with the propositions first when making the table 3. Follow PEMDAS, do the items in the parenthesis 4. Then do the rest 5. End with the compound proposition How many rows? 1. Let n = amount of simple propositions (p, q, r; single letter propositions) 2. 2n How much truth, and how much false for the simple parameters? Example Just follow this pattern. 1st proposition will always be half 2nd – so on, proposition will alternate and so on