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Mathematical Logic Short Notes
Gateforum
Discrete Mathematics
1 Propositional Logic
Propositions : A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both. We say that the truth value of a proposition is either true (T or 1) or false ( F or 0). New propositions, called compound propositions, are formed from existing propo- sitions using logical operators.
Logical Operators : Let p,q be two propositions, (a) negation: “not p”, ¬p, ∼ p, p′, ¯p (b) conjunction: “p and q”, p ∧ q, pq (c) disjunction: “p or q”, p ∨ q, p + q (d) exclusive or: “exactly one of p or q”, “p xor q”, p ⊕ q. (e) implication: “if p then q”, p → q. (f) biconditional: “p if and only if q”, p ↔ q, p iff q
Truth Table :
p q ¬p p ∧ q p ∨ q p ⊕ q p → q p ↔ q F F T F F F T T F T T F T T T F T F F F T T F F T T F T T F T T
Implication Operator : This is one of the most important operator in logic. Let p, q be two propositions. We say ”if p then q ”, written as p → q, is False only when p is true and q is false, otherwise true. p → q is true when either p is false or q is true. p → q ≡ p′^ + q
p → q can also be expressed as following :
- “if p, then q”
- “p implies q”
- “if p, q”
- “p only if q”
- “p is sufficient for q”
- “a sufficient condition for q is p” 7.“q if p”
- “q whenever p”
- “q when p”
- “q is necessary for p”
- “a necessary condition for p is q”
- “q follows from p”
- “q unless ¬p”
CONVERSE, CONTRAPOSITIVE, AND INVERSE :
For the compound statement p → q,
- q → p is the converse of p → q.
- ¬p → ¬q is the inverse of p → q.
- ¬q → ¬p is the contrapositive of p → q. A conditional statement is equivalent to its contrapositive. A conditional statement is NOT equivalent to its converse or inverse.
Precedence of Logical Operators :
¬ > ∧ > ∨ > → > ↔
Tautology, Contradiction, Contingency, Satisfiability : A compound proposition that is always true is called a tautology. A compound proposition that is always false is called a contradiction. A compound proposition that is neither a tautology nor a contradiction is called a contingency. A compound proposition that is either a tautology or a contingency is called Satisfiable. A compound proposition that is a contradiction, is called unsatisfi- able.
Propositional Equivalences : Compound propositions that have the same truth values in all possible cases are called logically equivalent. The compound propositions p and q are called logically equivalent, denoted by p ≡ q, if p ↔ q is a tautology, i.e. p → q and p ← q.
Some laws in logic :
Let p,q,r be any three propositions.
Existential quantifier : Let Q(x) be a predicate and D the domain of x. An existential statement is a statement of the form “∃x ∈ D such that Q(x).” It is defined to be true if, and only if, Q(x) is true for at least one x in D. It is false if, and only if, Q(x) is false for all x in D. Here ∃ is called the existential quantifier.
Negating Quantified Expressions : ¬∀xF ≡ ∃x¬F ¬∃xF ≡ ∀x¬F
Nested Quantifiers : Let Q(x, y) be some predicate where domain for variable x, y is D.
- ∀x∀yP (x, y) : It is true iff for every pair (x, y), Q(x, y) is true. (Where x, y ∈ D )
- ∀x∃yP (x, y) : It is true iff for every element x, there is some y such that Q(x, y) is true. (Where x, y ∈ D ). NOTE that for different elements x 1 , x2 ; y values could be different.
- ∃x∀yP (x, y) : It is true iff for some fixed element x, and for all elements y, Q(x, y) is true. (Where x, y ∈ D )
- ∃x∃yP (x, y) : It is true iff for some pair (x, y), Q(x, y) is true. (Where x, y ∈ D )
