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Mathematical Logic Short Notes: Propositional and Predicate Logic, Lecture notes of Discrete Mathematics

Logic notes in discrete mathematics

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2019/2020

Uploaded on 04/30/2020

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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,p0, ¯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, piff 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 pthen q”, written as pq, is False only when pis true and qis
false, otherwise true. pqis true when either pis false or qis true.
pqp0+q
1
pf3
pf4

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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 :

  1. “if p, then q”
  2. “p implies q”
  3. “if p, q”
  4. “p only if q”
  5. “p is sufficient for q”
  6. “a sufficient condition for q is p” 7.“q if p”
  7. “q whenever p”
  8. “q when p”
  9. “q is necessary for p”
  10. “a necessary condition for p is q”
  11. “q follows from p”
  12. “q unless ¬p”

CONVERSE, CONTRAPOSITIVE, AND INVERSE :

For the compound statement p → q,

  1. q → p is the converse of p → q.
  2. ¬p → ¬q is the inverse of p → q.
  3. ¬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.

  1. ∀x∀yP (x, y) : It is true iff for every pair (x, y), Q(x, y) is true. (Where x, y ∈ D )
  2. ∀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.
  3. ∃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 )
  4. ∃x∃yP (x, y) : It is true iff for some pair (x, y), Q(x, y) is true. (Where x, y ∈ D )