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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 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

• q if p

• q when p

• if p then q

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