Mathematical Language and Symbols (1) PDF

MATHEMATICAL LANGUAGE AND SYMBOLS 6.53 Chapter 2 LEARNING OBJECTIVES Discuss the language, Evaluate mathematical symbols, and conventions expressions correctly. used in mathematics. Explain the nature of Recognize that mathematics as a language. mathematics is a useful...

MATHEMATICAL LANGUAGE AND SYMBOLS 6.53 Chapter 2 LEARNING OBJECTIVES Discuss the language, Evaluate mathematical symbols, and conventions expressions correctly. used in mathematics. Explain the nature of Recognize that mathematics as a language. mathematics is a useful language. 2 KEYWORDS Expressions vs. Sentences Sets Logical Functions Connectives Relations Quantifiers Unary and Negation Binary Operations Free and Bound Variables Propositions Converse Inverse Contrapositive 3  ENGLISH NOUN SENTENCE (name given to object of interest) (must state a complete thought) SOMETIMES TRUE / PERSON PLACE THING TRUE (T) FALSE (F) SOMETIMES FALSE Carol Manila Dog The capital of the The capital of The dog is Philippines is the Philippines is black. Manila Makati MATHEMATICS EXPRESSION SENTENCE (name given to mathematical object (must state a complete thought) of interest) SOMETIMES TRUE / TRUE (T) FALSE (F) SOMETIMES FALSE NUMBER SET FUNCTION MATRIX ORDERED PAIR 1+1=2 1 + 1 = 11 x=1 {8} f(x) 1 4 (x,y) 8 −2 3 THE LANGUAGE OF MATHEMATICS Mathematical language is precise which means it is able to make very fine distinctions or definitions among a set of mathematical symbols. It is concise because a mathematician can express otherwise long expositions or sentences briefly by using the language of mathematics. The mathematical language is powerful, that is, one can express complex thoughts with relative ease.  EXPRESSIONS VS. SENTENCES A sentence must contain a complete thought. In the English language an ordinary sentence must contain a subject and a predicate. The subject contains a noun or a whole clause. “Manila” for example is a proper noun but is not in itself a sentence because it does not state a complete thought. Similarly, a mathematical sentence must state a complete thought. An expression is a name given to a mathematical object of interest. The term “1 + 2” is a mathematical expression but not a mathematical sentence. 7/1/20XX Pitch deck title 6  EXPRESSIONS VS. SENTENCES A mathematical expression is the mathematical analogue of an English noun. That is, it is a correct arrangement of mathematical symbols used to represent a mathematical object of interest (Burns, n.d). It does not state a complete thought and it does not make sense to ask if an expression is true or false. In mathematics areas, such as Algebra, the most common expressions are numbers, sets, and functions. 7/1/20XX Pitch deck title 7  EXPRESSIONS VS. SENTENCES A mathematical sentence is the mathematical analogue of an English sentence. That is, it is a correct arrangement of mathematical symbols that state a complete thought. Hence, it makes sense to ask if a sentence is true, false, sometimes true, or sometimes false. 7/1/20XX Pitch deck title 8 CARDINAL ORDINAL NOMINAL NUMBERS NUMBERS NUMBERS CARDINAL NUMBERS Used for counting and answer the question “how many?” Cardinal Numbers Examples The cardinality of a group represents the number of objects available in that group. 1. There are 6 clothes in the cupboard. 2. 4 cars are driving in a lane. 3. Anusha has 2 dogs and 1 cat as pets in her house. In the above three examples, the numbers 6, 4, 2 and 1 are the cardinal numbers. So basically it denotes the quantity of something, irrespective of its order. It defines the measure of the size of a set but does not take account of the order. ORDINAL NUMBERS Tell the position of a thing in terms of first, second, third, etc. Examples: 1. Anil came to 3rd position in a running competition. 2. The 6th chair is broken in a hall. If several objects are mentioned in a list, the order of the objects is defined by ordinal numbers. The adjective terms which are used to denote the order of something are 1st, 2nd, 3rd, 4th, 5th, 6th, and so on. Cardinal Numbers Ordinal Numbers They are used to denote the rank or position or order They are used for counting purposes of something or someone Examples: 1, 2, 3, 5, 6, 10, etc. Examples: 1st, 2nd, 5th, 6th, 9th, etc Question: ‘Where’ or ‘Which’ Question type: ‘How many?’ For example: Where is this located in the field? Or For example: How many balls are there? Which position does it hold in the ground? NOMINAL NUMBERS Used only as a name, or to identify something (not as an actual value or position). Examples: 1. The number at the back of Michael Jordan is “23”. 2. The postal zip code of Apalit is 2016. 3. The plate number of the car is EBX 528. It is not for representing the quantity or the position of an object. UNARY AND BINARY OPERATIONS  UNARY OPERATIONS The plus and minus signs may not mean addition or subtraction when they are attached before a single number. Instead, they are read as positive and negative signs. When written this way, they are called unary operations. 7/1/20XX  UNARY OPERATIONS The positive sign is very much like the addition operation but has a different meaning when attached to only one number. 7/1/20XX  UNARY OPERATIONS For example; +4 (read as ‘positive four’). It does not really mean ‘add four’. The value of four represented by the number 4 is considered as a single operand by the unary operator, ‘+’, and that operation produces a value of positive four. 7/1/20XX  UNARY OPERATIONS The negative, (or opposite), sign is a unary operator. Consider this expression: -4. Technically here, the negative sign operator accepts a value of four as its operand and produces a value of negative four. 7/1/20XX  UNARY OPERATIONS In summary, unary operations involve only one value. Examples: -5 sin x cos 45˚ π tan 3 7/1/20XX BINARY OPERATIONS When “+” and “-” can act on two operands, then it is called a binary operation. Consider this expression: 3 – (-2) in this expression, two operations are present using the symbol ‘-’ the first symbol (left most), is the binary subtraction operation. the other (right before the integer 2) is the unary negative sign operator. BINARY OPERATIONS An operation is binary if it takes two real numbers as arguments to produce another real number. BINARY OPERATIONS Examples: Addition. 4 + 5 = 9 Subtraction. 10 – 8 = 2 Multiplication. 4 x 6 = 24 PROPERTIES of Two Binary Operations, Addition and Multiplication, over the Set of Real Numbers PROPERTIES Closure of Binary Operations The product and the sum of any two real numbers is also a real number. PROPERTIES Commutativity of Binary Operations addition and multiplication of any two real numbers is commutative. PROPERTIES Associativity of Binary Operations given any three real numbers you may take away two and perform addition or multiplication as the case maybe and you will end with the same answer. PROPERTIES Distributivity of Binary Operations distributivity applies when multiplication is performed on a group of two numbers added or subtracted together. PROPERTIES Identity Elements of Binary Operations the identity is the number that you add or multiply to any real number and the result will be the same real number. PROPERTIES Inverses of Binary Operations the number that you add or multiply to get the identity element.  SOME FUNDAMENTALS OF LOGIC 7/1/20XX Pitch deck title 30  While “logic” may simply refer to valid reasoning in everyday life, it is also one of the oldest and most foundational branches of mathematics, often blurring the boundaries between mathematics and philosophy. 7/1/20XX Pitch deck title 31  Logic is the study of Truth and how we can obtain universal Truths through mathematical deduction. It is the most basic language of mathematics, and the underlying principle of proof. Mathematical logic and reasoning dates back many thousand years, to ancient Egyptian architects and Babylonian astronomers. Logical thinking also developed independently in India and China. 7/1/20XX Pitch deck title 32 PROPOSITION A proposition is a statement which is either True (T) or False (F). It must be one or the other and it cannot be both. A proposition is the basic building block of logic. It is defined as a declarative sentence that is either True or False, but not both. 7/1/20XX Pitch deck title 33 PROPOSITION Each of the following statements is a proposition: 1. The sun rises in the East and sets in the West. 2. 1 + 1 = 2 3. ‘b’ is a vowel. All of the above sentences are propositions, where the first two are Valid(True) and the third one is Invalid(False). Some sentences that do not have a truth value or may have more than one truth value are not propositions. 7/1/20XX Pitch deck title 34 TRUTH VALUE Truth value is the attribute of the proposition as to whether the proposition is true or false. 7/1/20XX Pitch deck title 35 TRUTH TABLE A table that shows the truth value of a compound statement for all possible truth values of its simple statements. 7/1/20XX Pitch deck title 36 NEGATION A statement is a negation of another if the word is not introduced in the negative statement. The action or logical operation of negating or making negative Let P be a proposition. The negation of P is “not P” or ¬𝑃. NEGATION P ¬𝑃 T F F T LOGICAL CONNECTIVES It is the mathematical equivalent of a conjunction in English. “and” ∧ “or” ∨ 7/1/20XX Pitch deck title 39 LOGICAL CONNECTIVES If two statements are joined like P and Q, denoted by 𝑃 ∧ 𝑄, then 𝑃 ∧ 𝑄 is a statement that is true if and only if both P and Q are true. The statement 𝑃 ∨ 𝑄 is true if and only if P is true or Q is true, and when they are both true. 7/1/20XX Pitch deck title 40 LOGICAL CONNECTIVES P Q 𝑃∧𝑄 𝑃∨𝑄 F F F F F T F T T F F T T T T T 7/1/20XX Pitch deck title 41 IMPLICATIONS A relationship between two propositions in which the second is a logical consequence of the first. Suppose P and Q are propositions. The proposition 𝑃 ⇒ 𝑄 (read as “if P, then Q”) is called an implication. 7/1/20XX Pitch deck title 42 IMPLICATIONS P is called the premise and Q is called the conclusion. “If it rains, then I bring my umbrella” is an implication. “If it rains” is P or the premise “I bring my umbrella” is Q or the conclusion. 7/1/20XX Pitch deck title 43 IMPLICATIONS P implies Q Q if P Q is implied by P Q only if P 7/1/20XX Pitch deck title 44 IMPLICATIONS P Q 𝑃⇒𝑄 F F T F T T T F F T T T 7/1/20XX Pitch deck title 45 IMPLICATIONS A more complicated form of implication is the bi implication or the biconditional denoted by the symbol ⇔. The statement 𝑃 ⟺ 𝑄 is true if and only if both P and Q are either both true or both false 7/1/20XX Pitch deck title 46 CONVERSE Suppose P and Q are propositions. Given the implication 𝑃 ⇒ 𝑄. Its Converse is Q ⇒ 𝑃. INVERSE Suppose P and Q are propositions. Given the implication 𝑃 ⇒ 𝑄. Its Inverse is ¬𝑃 ⇒ ¬𝑄. CONTRAPOSITIVE Suppose P and Q are propositions. Given the implication 𝑃 ⇒ 𝑄. Its Contrapositive is ¬𝑄 ⇒ ¬𝑃. CONVERSE, INVERSE, CONTRAPOSITIVE That is, Given: If P then Q. Inverse: If not P then not Q. Converse: If Q then P. Contrapositive: If not Q then not P. P Q 𝑃⇒𝑄 Inverse Converse Contrapositive ¬𝑃 ⇒ ¬𝑄 Q⇒𝑃 ¬𝑄 ⇒ ¬𝑃 F F T T T T F T T F F T T F F T T F T T T T T T  A conditional statement consists of two parts, a hypothesis/premise in the “if” clause and a conclusion in the “then” clause. For instance, “If it rains, then they cancel school.” "It rains" is the hypothesis/premise. "They cancel school" is the conclusion. To form the converse of the conditional statement, interchange the premise and the conclusion. The converse of "If it rains, then they cancel school" is "If they cancel school, then it rains." To form the inverse of the conditional statement, take the negation of both the premise and the conclusion. The inverse of “If it rains, then they cancel school” is “If it does not rain, then they do not cancel school.” To form the contrapositive of the conditional statement, interchange the premise and the conclusion of the inverse statement. The contrapositive of "If it rains, then they cancel school" is "If they do not cancel school, then it does not rain." Statement If two angles are congruent, then they have the same measure. Converse If two angles have the same measure, then they are congruent. If two angles are not congruent, then they do not have the same Inverse measure. If two angles do not have the same measure, then they are not Contrapositive congruent.  If a quadrilateral is a rectangle, then it has two pairs of Statement parallel sides. If a quadrilateral has two pairs of parallel sides, then it is a Converse rectangle. (FALSE!) If a quadrilateral is not a rectangle, then it does not have Inverse two pairs of parallel sides. (FALSE!) If a quadrilateral does not have two pairs of parallel sides, Contrapositive then it is not a rectangle. QUANTIFIERS Quantifiers are used to describe the variable(s) in a statement. Quantifiers are words, expressions, or phrases that indicate the number of elements that a statement pertains to. In mathematical logic, there are two quantifiers: 'there exists' and 'for all. UNIVERSAL QUANTIFIER Is usually written in the English language as “for all” or “for every”. EXISTENTIAL QUANTIFIER Is expressed in words as “there exists” or “for some” https://calcworkshop.com/reasoning-proof/conditional- statement/#:~:text=Example,as%20Math%20Planet%20accurately%20states. THANK YOU