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Mathematics and Statistics: Mathematical Logic

Topics are arranged as per SYLLABI published by Maharashtra State Board of Secondary and Higher Secondary Education.

A declarative (assertive) sentence which is either true or false is called statement in logic.

Sentence Statement or Proposition
Group of words having some meaning. Sentence is a statement, if its truth or falsity can be decided.
Types of Sentences:
  • Declarative or Assertive : Makes a statement or assertion
  • Interrogative : Asks a question
  • Imperative: Expresses a command
  • Exclamatory: Express strong feeling
Statement has only two values:
TRUE : denoted by T
FALSE: denoted by F
They are denoted by variables, p, q, r, s, etc. Imperative, Interrogative, Exclamatory are not statements.

  • If the statement is true, its truth value is "T"
  • If the statement is false, its truth value if "F"
  • A statement is either "true" or "false" but not both or does not have middle value between 'true' and 'false'. This is called law of excluded middle.
  • Ex: The statement "Mumbai is capital of Maharashtra state" is true and therefore it’s truth value is T.

Let us consider sentence Maths Logic; unless we know values of a,b,c and x we cannot say whether statement is true of false. An open sentence is a sentence whole truth can vary according to some conditions which are not stated in the sentence.

  • If a statement contains one or more connectives [ex: and], it is called 'composite' or 'compound' statement.
  • Simple statements (Mumbai is capital of Maharashtra state.) are used in forming a compound (Mumbai is capital of Maharashtra state and Gateway of India is in Mumbai.) statements are called 'prime components' or 'components' of the compound statement.

An open sentence with a quantifier becomes a statement and is called quantified statements. There are two types of quantifiers:

  • Universal Quantifier Maths Logic : All Values
  • Existential Quantifier Maths Logic : There exists
Statement "For every Maths Logic is non negative" can be written mathematically as Maths Logic

Conjunction Disjunction Negation Implication / Conditional Biconditional
AND (though) OR NOT IF.... THEN IF AND ONLY IF
Symbol: ∧ Symbol: ∨ Symbol: ∼ Symbol: Maths Logic or Maths Logic Symbol: Maths Logic or Maths Logic
Conjunction of p and q is denoted as p ∧ q Disjunction of p and q is denoted as p ∨ q If p is a statement ∼p is read as negation. p Maths Logic q is read as p implies q p Maths Logic q is read as p, if and only if q.
Components p and q are called conjuncts. Components p and q are called disjuncts Not used to combine two statements. p is antecedent and q is consequent. p and q are called as implicants.
Binary Connective Binary Connective Unary Connective Binary Connective Binary Connective
Note: In some books Negation is referred by symbol: Maths Logic

The truth table of a compound statement shows all the corresponding truth values of the compound statement for all the possible truth values of each prime statement of the compound statement.

A statement pattern is formed by using statements and one or more logic connectives.

Two statements are defined to be logically equivalent if and only if their truth values are identical. Logical equivalence of p and q is expressed as p ≡ q.
Tautology: A statement which is always TRUE.
Contradiction: A statement which is always FALSE. It is also referred as Fallacy.
Contingency: A statement which is neither tautology nor contradiction.
Duality: If A is any statement involving the connectives ∼, ∧, ∨ Then the dual of A is a statement pattern obtained from A by replacing ∧ with ∨ (or by replacing ∨ with ∧). Duel of A is denoted by A
Negation of compound Statement: is obtained by just inserting 'NOT' at the appropriate place in the statement.
Contrapositive:The statement ∼ q Maths Logic ∼ p is called contrapositive of p Maths Logic q . Here order as well as sign of each prime statement is changed.
Converse: The statement q Maths Logic p is called as the converse of p Maths Logic q. Here order of the prime statement is changed.
Inverse: The statement ∼p Maths Logic ∼ q is called inverse of p Maths Logic q. Sign of each prime statement is changed.

Idempotent law: p ∧ p Maths Logic p
p ∨ p Maths Logic p
Associative law: (p ∨ q) ∨ r = p ∨ (q ∨ r)
(p ∧ q) ∧ r = p ∧ (q ∧ r)
Commutative law: p ∨ q = q ∨ q
p ∧ q = q ∧ p
Distributive law: p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r)
Identity law: [The condition that is always true is denoted by T and the condition that is always false is denoted by F]
p ∧ T Maths Logic p
p ∨ F Maths Logic p
Complement Law: ('t' denotes tautology and 'c' denotes contradiction)
p ∨ ∼ p = t
p ∧ ∼ p = c
∼(∼ p) = p
t = c
c = t
Involution Law: ∼(∼ p) ≡ p
DeMorgan’s laws:

  • Negation of conjunction of two statements is logically equivalent to the disjunction of their negation. It is represented as ∼(p ∧ q) ≡ ∼p ∨ ∼q
  • Negation of disjunction of two statements is logically equivalent to the conjunction of their negation. It is represented as ∼(p∨q) ≡ ∼p ∧ ∼q
Difference between converse, contrapositive, contradiction:
Converse of = pMaths Logicq is qMaths Logicp
Contrapositive of pMaths Logicq is ∼qMaths Logic∼p
Contradiction of p is ∼p

Mathematical Logic has two values either true or false. Similar situation exists in electrical switch. It has two states 'on' (closed) and 'off' (open). Basic series Network and Basic Parallel network serves as building blocks for various types of networks:
Basic Series Network:

  • It is denoted by A ∧ B or A.B
  • Current will flow only when both A and B are in 'ON' (Closed) state.
  • It is often called as AND circuit.

Basic Parallel Network:
  • It is denoted by A ∨ B or A + B
  • Current will flow when anyone A or B is in 'ON' (Closed) state.
  • It is often called as OR circuit.
The arrangement of wires and switches that can be constructed by repeated use of series and parallel circuit is called Boolean Switching Circuit. The behaviour or state of Boolean switching circuit can be obtained by constructing a table which is similar to the truth table of a compound proposition.
Any combination of switches using the connective ∨ and ∧ is called a Boolean Polynomial.

  • The negation of conjunction of two statements is logically equivalent to the disjunction of their negations.
  • Negation of negation of a statement is logically equivalent to the statement itself.
  • Negation of disjunction of two statements is logically equivalent to the conjunction of their negation.
  • Open sentences are NOT considered as statements in Logic.
  • The axiomatic approach to logic was first propounded by an English Philosopher and Mathematician George Boole. Hence it is called Boolean Logic or Symbolic Logic.
  • Methods used to check validity of statements are:
    • Direct Method
    • Contra-positive Method
    • Method of Contradiction
    • Using a counter example
  • A mathematically acceptable statement is a sentence which is either true or false.