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Mathematical Reasoning

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Mathematical Reasoning PDF Notes, Important Questions and Formulas

Key Concepts

  1. There are two types of reasoning- deductive and inductive. Deductive reasoning was developed by Aristotle, Thales, and Pythagoras in the classical Period (600-300 B.C.).
  2. In deduction, given a statement to be proven, often called a conjecture or a theorem, valid deductive steps are derived and a proof may or may not be established. Deduction is the application of a general case to a particular case.
  3. Inductive reasoning depends on working with each case and developing a conjecture by observing incidence till each and every case is observed.
  4. Deductive approach is known as the ‘top-down approach’. The given theorem is narrowed down to a specific hypothesis, and then to an observation. Finally the hypothesis is tested with specific data to get the confirmation (or not) of the original theory.                                                                                                                                                           
  5. Mathematical reasoning is based on deductive reasoning.
    The classic example of deductive reasoning given by Aristotle is
    • All men are mortal.
    • Socrates is a man.
    • Socrates is mortal.
  6. The basic unit involved in reasoning is the mathematical statement.

  7. A sentence is called a mathematically acceptable statement if it is either true or false but not both. A sentence which is both true and false simultaneously is called a paradox.
  8. Sentences which involve tomorrow, yesterday, here, there etc., i.e. variables are not statements.
  9. Sentences which express a request, a command or is simply a question are not statements.
  10. The denial of a statement is called the negation of the statement.
  11. Two or more statements joined by words such as ‘and’, ‘or’ are called compound statements. Each statement is called a component statement, where ‘and’ ‘or’ are the connecting words.
  12. An ‘AND’ statement is true if each of the component statements is true, and it is false even if one of the component statements is false.
  13. An ‘OR’ statement is true when even one of its components is true; and it is false only when all its components are false.
  14. The word ‘OR’ can be used in two ways (i) Inclusive OR (ii) Exclusive OR. If only one of the two options is possible then the OR used is an exclusive OR. If any one of the two options or both the options are possible then the OR used is an inclusive OR.
  15. ‘There exists’ (Ǝ) and ‘For all’ (⩝) are called quantifiers.
  16. A statement with the quantifier ‘There exists’ is true, if is true for at least case.
  17. If p and q are two statement then a statement of the form ‘If p then q is known as a conditional statement. In symbolic form, p implies q is denoted by p ⇒ q
  18. The conditional statement p ⇒ q can be expressed in various other forms such as:                                                          (i) q if p (ii) p only if q (iii) p is sufficient q (iv) q is necessary for p.  
  19. A statement formed by the combination of two statements of the form if p then q and q then p and is called and only if implication. And it is denoted p ⇔ q. It is called biconditional statement.
  20. Contra positive and converse can be obtained an ‘if then’ statement. The contrapositive of a statement p ⇒ q is the statement ~ q ⇒ ~ p; the converse of a statement p ⇒ q is the statement q ⇒ p
  21. Truth values of various statements                                                                                                                               

    p

    q

    P and q

    P or q

     P ⇒ q

    T

    T

    T

    T

    T

    T

    F

    F

    T

    F

    F

    T

    F

    T

    T

    F

    F

    F

    F

    T

                              
  22. To prove the truth of an ‘if p- then q statement’ there are two ways, Assume p is true and prove q is true. This is called the direct method.                                                                                                                                                                                                                         Or                                                                                                          Assume that q is false and prove p is false. This is called the contrapositive method.
  23. To prove the truth of ‘p if and only if q’ statement, we must prove two things, one is that the truth of p implies the truth of q and the second is that the truth of q implies the truth of p.
  24. The following methods are used to check the validity of statements:
  1. Direct method
  2. Contrapositive method
  3. Method of contradiction
  4. Using a counter example 25.To check whether a statement p is true, we assume that it is not true, i.e. ~ p is true. Then we arrive at some result which contradicts our assumption.
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