Maharashtra State Board · Class 12 · Mathematics · Part I Chapter 1

Mathematical Logic — Formula Sheet

Board Formulas

18 formulas

1.Negation
$\sim p is T ⟺ p is F$
$p$ : A statement (truth value T or F) · $\sim p$ : Negation of p, read 'not p'
Negation reverses the truth value. In words, add 'not' or 'It is false that…' to the statement.
2.Conjunction (and)
$p \land q is T ⟺ p and q are both T$
$p, q$ : Component statements
Words such as 'and', 'but', 'yet', 'still' all give a conjunction. One false component makes the whole statement false.
3.Disjunction (or)
$p \lor q is F ⟺ p and q are both F$
Mathematical 'or' is inclusive: p ∨ q is true when at least one of p, q is true, including when both are.
4.Conditional (if … then)★
$p \to q is F ⟺ p is T and q is F$
p is the antecedent, q the consequent. Also read as 'p only if q', 'q if p', 'p is sufficient for q', 'q is necessary for p'.
5.Biconditional (if and only if)
$p \leftrightarrow q is T ⟺ p, q have the same truth value$
Read 'p if and only if q' or 'p is necessary and sufficient for q'.
6.Number of Rows in a Truth Table
$rows = 2^{n}$
$n$ : Number of simple statements in the pattern
Two statements need 4 rows, three statements need 8 rows. Use the order TT, TF, FT, FF for two statements.
7.Tautology and Contradiction★
$p \lor \sim p \equiv T, p \land \sim p \equiv F$
Tautology: last column all T. Contradiction: last column all F. Contingency: a mixture of T and F.
8.Conditional as a Disjunction
$p \to q \equiv\sim p \lor q$
The most useful equivalence of the chapter. Use it to remove → before applying De Morgan's laws or finding a dual.
9.Converse, Inverse and Contrapositive
$Converse: q \to p, Inverse: \sim p \to\sim q, Contrapositive: \sim q \to\sim p$
All three are formed from the conditional p → q. Write each one in words when the statement is given in words.
10.Contrapositive Equivalence★
$p \to q \equiv\sim q \to\sim p, q \to p \equiv\sim p \to\sim q$
A conditional and its contrapositive always have the same truth table. So do the converse and the inverse.
11.Biconditional as Two Conditionals
$p \leftrightarrow q \equiv (p \to q) \land (q \to p)$
Use this to prove a biconditional or to find its negation.
12.Duality
$Dual of p \lor (q \land r) is p \land (q \lor r)$
Replace ∧ by ∨, ∨ by ∧, t by c and c by t (in duality the textbook writes t for a tautology and c for a contradiction). Negation signs and the statements stay as they are. Remove → and ↔ first.
13.De Morgan's Laws★
$\sim (p \land q) \equiv\sim p \lor \sim q, \sim (p \lor q) \equiv\sim p \land \sim q$
Negate each component AND switch the connective. 'Not (A and B)' means 'not A or not B'.
14.Negation of a Conditional★
$\sim (p \to q) \equiv p \land \sim q$
The negation of 'If p then q' is 'p and not q' — it is NOT another if-then sentence.
15.Negation of a Biconditional
$\sim (p \leftrightarrow q) \equiv (p \land \sim q) \lor (q \land \sim p)$
Follows from p ↔ q ≡ (p → q) ∧ (q → p), then De Morgan and the negation of each conditional.
16.Negation of Quantified Statements★
$\sim [\forall x, p (x)] \equiv \exists x such that \sim p (x), \sim [\exists x such that p (x)] \equiv \forall x, \sim p (x)$
$\forall$ : Universal quantifier: 'for all', 'for every' · $\exists$ : Existential quantifier: 'there exists', 'for some' · $p (x)$ : Open sentence in the variable x
'For every' becomes 'there exists' and vice versa; the open sentence p(x) is negated. The set (N, Z, R…) does not change.
17.Distributive Laws
$p \land (q \lor r) \equiv (p \land q) \lor (p \land r), p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$
Each connective distributes over the other. Read right to left, they let you take a common statement outside a bracket.
18.Identity and Involution Laws
$p \land T \equiv p, p \lor F \equiv p, p \lor T \equiv T, p \land F \equiv F, \sim (\sim p) \equiv p$
T is any tautology and F any contradiction. These laws finish most simplifications after a distributive step.