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Mathematical Reasoning

  • A sentence is called a mathematically acceptable statement if it is either true or false but not both.

  • A sentence is neither imperative nor interrogative nor exclamatory.

  • A declarative sentence containing variables is an open statement if it becomes a statement when the variables are replaced by some definite values.

  • A compound statement is a statement which is made up of two or more statements. Each of this statement is termed to be a compound statement.

  • The compound statements are combined by the word “and” (^) the resulting statement is called a conjunction denoted as p ∧ q.

  • The compound statement with “And” is true if all its component statements are true.

  • The following truth table shows the truth values of p ∧ q ( p and q) and q ∧ p ( q and p):

Truth Table (p  q, q  p)

p

q

p ∧ q

q ∧ p

T

T

T

T

T

F

F

F

F

T

F

F

F

F

F

F

Rule: p ∧ q is true only when p and q are true

  • Compound statements p and q are combined by the connective ‘OR’ (∨) then the compound statement denoted as p ∨ q so formed is called a disjunction.??

Truth Table (p v q, q v p)

p

q

q

p

T

T

T

T

T

F

T

T

F

T

T

T

F

F

F

F

Rule: p  q is false only when both p and q are false.

  • The denial of a statement is called the negation of the statement. The truth table for the same is given below:

Truth Table (~p)

p

~p

~ (~p)

T

F

T

F

T

F

Rule: ~ is true only when p is false

  • Negation is not a binary operation, it is a unary operation i.e. a modifier.

  • There are three types of implications: 

  • “If ….... then” 

  • “Only if”

  • “If and only if”

  • “If …. then” type of compound statement is called conditional statement. The statement ‘if p then q’ is denoted by p → q or by p ⇒ q. p → q also means:

  • p is sufficient for q

  • q is necessary for p

  • p only if q

  • p leads to q

  • q if p

  • q when p

  • if p then q

  • Truth table for p → q
  • Truth Table (p  q, q  p)

    p

    q

     q

     p

    T

    T

    T

    T

    T

    F

    F

    T

    F

    T

    T

    F

    F

    F

    T

    T

    Rule: p  q is false only when p is true and q is false.

  • If and only if type of compound statement is called biconditional or equivalence or double conditional. represnted as p ⇔ q or p ↔ q, it means

  • p is a necessary and sufficient condition for q

  • q is a necessary and sufficient condition for p

  • If p then q and if q then p

  • q if and only if p

  • Truth table for p ↔ q or q ↔ p

Truth Table (p  q, q  p)

p

q

p ↔ q

q ↔ p

T

T

T

T

T

F

F

F

F

T

F

F

F

F

T

T

Rule: p  q is true only when both p and q have the same truth value.

  • Contrapositive of p → q is q → p.

  • Converse of p → q is q → p.

  • Truth table for p → q

Truth Table (p  q)

 

p

q

p → q

 ~ ~p (Contrapositive)

q ↔ p (Converse)

T

T

T

T

T

T

F

F

F

T

F

T

T

T

F

F

F

T

T

T

  • The compound statements which are true for any truth value of their components are called tautologies.

  • Truth table for a tautology ‘p ∨ ~p”?, p being a logical statement

Truth Table (∨ ~p)

p

~p

∨ ~p

T

F

T

F

T

T

  • ?The negation of tautology is a fallacy or a contradiction. The truth table for ‘p ∧ p” which is fallacy?, p being a logical statement is given below

Truth Table (p ∧ ~p)

p

~p

p ∧ ~p

T

F

F

F

T

F

  • Important points on tautology and fallacy:

  • p ∨ q is true iff at least one of p and q is true

  • p ∧ q is true iff both p and q are true

  • A tautology is always true

  • A fallacy is always false.

  • ?Statements satisfy the following laws:

  • Idempotent Laws: If p is any statement then p ∨ p = p  and p ∧ p = p

  • Associative Laws: If p, q, r are any three statements, then p ∨ (q ∨ r) = (p ∨ q) ∨ r  and p ∧ (q ∧ r) = (p ∧ q) ∧ r

  • Commutative Laws: If p, q are any two statements, then p ∨ q = q ∨ p and p ∧ q = q ∧ p

  • Distributive Laws: If p, q, r are any three statements then p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r) and  (q ∧ r) = (p  q) ∧ (p ∨ r)

  • Identity Laws: If p is any statement, t is tautology and c is a contradiction, then p ∨ t = t, p ∧ t = p, p ∨ c = p and p ∧ c = c

  • Complement Laws: If t is tautology, c is a contradiction and p is any statement then p ∨ (~p) = t, p ∧ (~p) = c, ~t = c and ~c = t

  • Involution Law: If p is any statement, then ~(~p) = p

  • De-Morgan’s Law: If p and q are two statements then ~(p ∨ q) = (~p) ∧ (~q) and ~(p ∧ q) = (~p) ∨ (~q)

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