MATM111

• MATHEMATICS – It is formal system thought for recognizing, clarifying, and exploiting of patterns.
• MATH IS ABOUT

  Numbers, Symbols, Notations

  Operations, Equations, Functions

  Process and thingification of processes
• WHERE IS MATH?

  Math is every where
• MATH IS FOR…

   To help us unravel the puzzle of nature, a useful way to think about nature

   Organize patterns and regularities as well as irregularities

   To help us control weather and epidemics

   Provides new questions to think about
• MATHEMATICS IS DONE…

   With Curiosity

   With eagerness for seeking patterns and generalizations

   With desire to know the truth

   With trial and error

   Without fear of facing more questions and problems
• WHO USES MATHEMATICS?

  Mathematicians: Pure and Applied

  Scientist: Natural and Social

  Practically EVERYONE
• PATTERNS – Are regular, repeated or recurring forms or designs.

  SEQUENCE- It is an enumerated collection of objects in which repetition are allowed.

  FIBONNACI SEQUENCE- It is a series of number in which the next number is found by adding up the two previous terms before it.

  LEONARDO PISANO – Fibonacci Sequence named after
• WHY DO WE NEED TO KNOW THE LANGUAGE OF MATHS?

  To be able to understand the idea or concepts of Math
• CHARACTERISTICS OF THE LANGUAGE OF MATHEMATICS

   PRECISE

   CONCISE

   POWERFUL

  COMPONENTS OF LANGUAGE OF MATHEMATICS

   VOCABULARY

   GRAMMAR
• ARISTOTLE – “Father of Logic”

  LOGIC- The science or study of the principles and techniques of correct reasoning.

  LOGOS- It is a Greek word which means speech and reasoning.

  PROPOSITION- It is a declarative statement which is true or false, but not both; and is the foundation of logic.
• SIMPLE PROPOSITION

   It expresses a single complete thought

   Also, statements which cannot be broken down without a loss in meaning
• COMPOUND PROPOSITION

   A proposition formed by combining two or more simple statements in a certain ways.

   Two connectives used for this purpose are the words “are” and “or”

   Statements which can be broken down without a change in meaning
• CONNECTIVES – symbols to form compound proposition

   Not – (~, - )

   And – ( ^ )

   Or – ( v )

   If, Then (→)

   If (if and only if ) ↔
• TRUTH TABLE – a tabular representation of truth values.

  p – It is called the hypothesis sometimes called Antecedent.

  q – It is called the conclusion sometimes called Consequent.

  Mono Set – A set with elements means “one or single”.

  Null Set – A set that has no elements.
• NEGATION (~, -)

  • The contradiction or denial of something.
  • If the statement is true make it false.
  • If the statement is false make it true.

  
• CONJUNCTION (and / ^)

  • If both p and q are true, then the conjunction is TRUE, otherwise it is FALSE generally expressed as “and”.
  • Words such as “but”, “however, and “nevertheless” are also used as conjunctions.

  
• DISJUNCTION (or / v)

   If both p and q are false, then the disjunction is FALSE, otherwise it is TRUE.

  
• CONDITIONAL (→)

  • “logical implication”
  • If, then statement p → q

  
• BICONDITIONAL (↔)

  • TRUE if both operands are true; and also TRUE if both are false, same values TRUE, otherwise it is FALSE.

  
• CONVERSE (q → p)

  -“If two angles are congruent, then they are

  vertical angles”

  INVERSE (~p → ~q)

  -“If two angles are not vertical angles, then

  they are not congruent”.

  CONTRAPOSITIVE (~q → ~p)

  -“If two angles are not congruent, then they

  are not vertical angles”.
• TAUTOLOGY – A statement where every

  entry in the last column using truth table is

  TRUE.

  CONTRADICTION – A statement where

  every entry in the last column using truth table

  is FALSE.

  CONTIGENCY – A statement that is neither a

  tautology or a contradiction. Mix of true or

  false.
• PROBLEM – Is a situation that confronts the

  learner that requires resolution, and for which

  the path to the answer is not immediately

  known.

  DRILL/EXERCISE – A situation that

  requires resolution but the method is clear and

  the way to the answer is easily seen.

  PROBLEM SOLVING – There is an obstacle

  that prevents one from seeing a clear path to

  the answer.

  CONJECTURE – An educated guess based

  on repeated observations of a particular

  pattern or process.
• DEDUCTIVE REASONING- Is a

  process of reaching to conclusion

  (conjecture) by applying a general

  assumptions, procedures or principles.

  (General to Specific).

   INDUCTIVE REASONING – Is a

  process of reaching a general

  conclusion (conjecture) by examining

  specific examples. (Specific to

  General).
• Polya’s Problem Solving Strategy – named after

  George Polya (1887 – 1985)

  4 Steps in Problem Solving Strategy

  1. Preparation: Understand the Problem
  2. Thinking Time: Devise a Plan
  3. Insight: Carry out the Plan
  4. Verification: Review the Solution