Logical Reasoning

Proof
1. Direct Proof
  1. outline
  2. Reason to use
    1. Its a straightforward method, suitable for proving easier statement
2. Contrapositive Proof
  1. Reason to use
    1. 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
  2. Outline
    1. If p, then q
    2. Suppose ~q
    3. Therefore ~p
3. Contradiction Proof
  1. Reason to use
    1. used in when the statement cannot be proven using both direct proof and contrapositive proof method
  2. 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.
  3. Outline depends on the statement since contradiction method can both be used to prove a statement or a conditional statement
4. Mathematical Induction
  1. Reason to use
    1. Used whenever the statement requires proving a certain sequence such as Fibonacci's sequence
  2. Outline
    1. Prove that first statement is true
    2. assume that n=k is true
    3. prove than n=k+1 is true
    4. conclude every S is true
Logic
1. Systematic way of thinking that allows us to deduce new information and to examine the meaning of sentence
Statement
1. Sentence or a mathematical expression that is either definitely true (T) or false (F), but not both
2. Biconditional statement
3. Conditional statement
  1. If p then q
    1. Direct proof
    2. Indirect proof
      1. Contrapositive
      2. Contradiction
    3. Mathematical induction
4. Types of statement
  1. Known to be true ( theorem & preposition)
  2. Truth unknown (conjectures)
  3. Known to be false
Real Number
1. Rational Number
  1. Integer
    1. Negative Integer
    2. Zero
    3. Positive Integer
      1. Prime Number
      2. Composite Number
    4. Symbol - Z
  2. Non Integer
  3. Symbol - Q
2. Irrational Number
3. Symbol - R
Complex Number
Natural Number
1. Not include number '0'
Whole Number
1. Include number '0'
Truth Table
1. Display relationship between truth values of statements
2. Logical possibilities = 2^n
  1. Example : 2^2=4
3. Connective
  1. Negation
    1. not
  2. Conjuction
    1. and
  3. Disjunction
    1. or
  4. Implication
    1. if p, then q
  5. Equivalence
    1. if and only if
Theorem
1. statement that is true and has been true
Known facts and rules
1. De Morgan's law
2. Parity
  1. Two integers have same parity if both even or both odd. otherwise, they have opposite parity
    1. Addition and substraction
    2. Multiplication
3. n = Odd number
4. n = Even number
Quantifiers
1. Universal
  1. For all
  2. For every
2. Existential
  1. There exist a
  2. There is a
  3. There is at least 1

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Logical Reasoning

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