Mathematical Reasoning

Validating Statements


  • Tautology (t) :- The compound statement which contain only T in the last column of their truth table is called tautology, which if denoted by t.
    Eg:- (p ∨ q) ∨ (∼p ∨ ∼q)
  • Contradiction (c):- The compound statement which contain only F in the last column of their truth table is called contradiction which is denoted by c.
    Eg: [(p → q) → p] ∧ ∼ p
  • Rule 1 : If p and q are mathematical statements, then in order to show that the statement "p and q" is true, the following steps are followed.
       Step-1 Show that the statement p is true.
       Step-2 Show that the statement q is true.
  • Rule 2 : Statements with "Or"
    If p and q are mathematical statements, then in order to show that the statement "p or q" is true, one must consider the following.
       Case 1 By assuming that p is false, show that q must be true.
       Case 2 By assuming that q is false, show that p must be true.
  • Rule 3 : Statements with "If-then"
    In order to prove the statement "if p then q" we need to show that any one of the following case is true.
       Case 1 By assuming that p is true, prove that q must be true. (Direct method)
       Case 2 By assuming that q is false, prove that p must be false. (Contrapositive method)
  • Rule 4 : Statements with "if and only if"
    In order to prove that statement "p if and only if q", we need to show
       (i) If p is true, then q is true and (ii) if q is true, then p is true.

Validating Statements Part-1

Validating Statements Part-2

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

        Algebra of statements

  • commutative laws : p ∨ q ≅ q ∨ p
                                 p ∧ q ≅ q ∧ p
  • Associative laws : p ∨ (q ∨ r) ≅ (p ∨ q) ∨ r
                               p ∧ (q ∧ r) ≅ (p ∧ q) ∧ r
  • Distributive laws : p ∨ (q ∧ r) ≅ (p ∨ q) ∧ (p ∨ r)
                               p ∧ (q ∨ r) ≅ (p ∧ q) ∨ (p ∧ r)
  • Demorgan's laws : ∼ (p ∨ q) ≅ ∼p ∧ ∼ q 
                                ∼ (p ∧ q) ≅ ∼p ∨ ∼ q
                                ∼ (p ⇒ q) ≅ p ∧ ∼ q
                                ∼ (p ⇔ q) ≅ (p ∧ ∼ q) ∨ (q ∧ ∼ p)
  • Complement laws : ∼ (∼p) ≅ p
                                p ∨ (∼p) ≅ t
                                p ∧ (∼p) ≅ c
                                ∼t ≅ c
                                ∼c ≅ t
  • Idempotent laws : p ∨ p ≅ p
                               p ∧ p ≅ p
  • Identify laws : p ∨ t ≅ t
                         p ∧ t ≅ p
                         p ∨ c ≅ p
                         p ∧ c ≅ c