Mathematical-Language-and-Symbols.pdf

Mathematical Language and Symbols “Mathematics is the language of Sciences, Business, Economics, Music, Architecture, Arts and even Politics.”  Mathematical Language is classified as a language which used to communicate through numbers, symbols, sets, operations, functions,...

Mathematical Language and Symbols “Mathematics is the language of Sciences, Business, Economics, Music, Architecture, Arts and even Politics.”  Mathematical Language is classified as a language which used to communicate through numbers, symbols, sets, operations, functions, equations, abstraction, and linearity, complexity of language, logic, coding and decoding information. It is also a system used to express, communicate and share mathematical information.  Mathematics has symbols that only those who study it understand these symbols. Some of these symbols are: Characteristics of the Mathematical Language is:  Precise, it is able to make very fine distinction or definitions.  Concise or brief i.e., if someone can say things in long exposition or sentences, mathematics can say it briefly.  Powerful i.e., one can express complex thoughts with relatively ease. Expression vs Sentence  An expression is the mathematical analogue of an English noun; it is a correct arrangement of mathematical symbols used to represent a mathematical object of interest. Expression vs Sentence  A mathematical sentence is the analogue of an English sentence; it is a correct arrangement of mathematical symbols that states a complete thought. For example “The sum of any two real number is also a real number” Elementary Logic: connectives, negation, quantifiers, variables  Logic is defined as “The study of truths based completely on the meanings of the terms they contain.” It includes the act of reasoning by humans in order to form thoughts and opinions, as well as classifications and judgments. Some forms of logic can also be performed by computers and even animals.  Mathematical Logic is defined as a method of reasoning, provides rules, and techniques to determine whether the arguments are valid.  The use of logic illustrates the importance of precision and conciseness in communicating mathematics. Proposition A proposition is a declarative sentence that is either true or false but not both. It is made in people’s statements many times. Propositions can be written as P is an example variable to use: Propositions  Examples: Each of the following statement is a proposition, some are true and some are false. Can you tell which are true, and which are false? If it false, state why. a) P: 9 is a prime number. b) Q: PhilSCA-Main Campus is located in Makati City. c) R: The sum of 3 and 8 is 11. d) S: 10 < -3. The following are not proposition a)P: What is your name? b)Q: The sum of two numbers. c)R: x is greater than 3.  It is an interrogative sentence.  It does not state a complete thought. So, we cannot say whether it is true or false. x is a variable representing a number. But why is it not a proposition? It is because specific value of x is not given so we cannot say whether it is true or false.  Compound Statement  It is combination of two or more propositional statements using logical connectives. These logical connectives are as follow: ❖Conjunction ❖Disjunction ❖Negation ❖Implication ❖Biconditional Conjunction– a conjunction is statement formed by joining statement using the word “and”. ❑𝑷: √2 is a rational number. ❑Q: 6 is an even number. ❑𝑷 ∧ 𝐐: √2 is a rational number and 6 is an even number. Disjunction – a disjunction is statement formed by joining statement using the word “or”. ❑𝑷: √2 is a rational number. ❑Q: 6 is an even number. ❑𝑷 ∨ 𝐐: √2 is a rational number or 6 is an even number. Negation– the negation of 𝑷, written ¬𝑷, is a statement obtained by negating statement 𝑷. ❑𝑷: √2 is a rational number. ❑¬𝑷: √2 is not a rational number  Implication – is a conditional statement and written as “if and then” where the 1st statement is called the premise while the 2nd statement is called the conclusion. ❑𝑷: √2 is a rational number. ❑Q: 6 is an even number. ❑𝑷→𝐐: If √2 is a rational number, then 6 is an even number. Other implication statements of 𝑷→Q If Q then P Biconditional – the statement “ P if and only if Q” is called the biconditional or bi- implication of P and Q and written as 𝑷↔Q ❑𝑷: √2 is a rational number. ❑Q: 6 is an even number. ❑𝑷 ↔𝐐: √2 is a rational number if and only if 6 is an even number.  Truth Tables for compound Statements A truth table is a table that shows the truth value of a compound statement for all possible truth values of its simple statement. Conjunction Disjunction Negation Implication Other Implication Biconditional