Logical Reasoning

Logical Reasoning

1. Proof

   • Direct Proof
     • outline
     • Reason to use
       • Its a straightforward method, suitable for proving easier statement
   • Contrapositive Proof
     • Reason to use
       • Certain cases can be both proved using contrapositive proof and direct proof, however, for more complex proving statement, its preferable to use this method as it is easier
     • Outline
       • If p, then q
       • Suppose ~q
       • Therefore ~p
   • Contradiction Proof
     • Reason to use
       • used in when the statement cannot be proven using both direct proof and contrapositive proof method
     • The proof by contradiction is grounded in the fact that any proposition must be either true or false, but not both true or false at the same time.
     • Outline depends on the statement since contradiction method can both be used to prove a statement or a conditional statement
   • Mathematical Induction
     • Reason to use
       • Used whenever the statement requires proving a certain sequence such as Fibonacci's sequence
     • Outline
       • Prove that first statement is true
       • assume that n=k is true
       • prove than n=k+1 is true
       • conclude every S is true

2. Logic

   • Systematic way of thinking that allows us to deduce new information and to examine the meaning of sentence

3. Statement

   • Sentence or a mathematical expression that is either definitely true (T) or false (F), but not both
   • Biconditional statement
   • Conditional statement
     • If p then q
       • Direct proof
       • Indirect proof
         • Contrapositive
         • Contradiction
       • Mathematical induction
   • Types of statement
     • Known to be true ( theorem & preposition)
     • Truth unknown (conjectures)
     • Known to be false

4. Real Number

   • Rational Number
     • Integer
       • Negative Integer
       • Zero
       • Positive Integer
         • Prime Number
         • Composite Number
       • Symbol - Z
     • Non Integer
     • Symbol - Q
   • Irrational Number
   • Symbol - R

5. Complex Number

6. Natural Number

   • Not include number '0'

7. Whole Number

   • Include number '0'

8. Truth Table

   • Display relationship between truth values of statements
   • Logical possibilities = 2^n
     • Example : 2^2=4
   • Connective
     • Negation
       • not
     • Conjuction
       • and
     • Disjunction
       • or
     • Implication
       • if p, then q
     • Equivalence
       • if and only if

9. Theorem

   • statement that is true and has been true

10. Known facts and rules

    • De Morgan's law
    • Parity
      • Two integers have same parity if both even or both odd. otherwise, they have opposite parity
        • Addition and substraction
        • Multiplication
    • n = Odd number
    • n = Even number

11. Quantifiers

    • Universal
      • For all
      • For every
    • Existential
      • There exist a
      • There is a
      • There is at least 1

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