Logic
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This document defines key concepts in logic and propositions: 1) A proposition is a statement that is either true or false, such as "Dogs have six legs" or "2+3=5". 2) The truth value of a proposition is either 1 (true) or 0 (false). 3) Two propositions are equivalent if they have the same truth value. The negation of a proposition, denoted by p', is its opposite truth value. 4) A compound proposition combines two or more simple propositions using logical connectives like "and", "or", or "if/then".
Logic
- 1. Logic
- 2. Propositions • An assertion which is definitely either true or false is called a proposition, shown by the letters p, q, r, s, …. • Examples: • Dogs with six legs. • 2+3=5 • Come here. • Who is that? • The triangle is negative.
- 3. Truth values of proposition • The truth and falsity of a proposition is called its truth value. The numbers 1 and 0 are used as the truth values of the true and false propositions respectively. p true 1 p false 0
- 4. p q 1 1 1 1 q 0 1 0 p 1 0 1 0 q 0 0 0
- 5. Equivalent propositions • Two propositions p and q have the same truth value they are said to be equivalent propositions, denoted as
- 6. Negation of a proposition • The negation of a proposition p is denoted by p’ . The following table provides some information about negation. p p’ Some symbols negation 1 0 = ≠ 0 1 > ≤ ≥ <
- 7. Compound Propositions • A proposition which is formed two or more propositions by using connective words is called a compound proposition. Connective Words: And Or Then If and only if
- 8. Connective Symbol Name and ∧ conjunction or ∨ Disjunction If …. then …. => Implication (implies) If and only if Equivalence
- 9. Conjunction ∧ • A compound proposition of p and q formed using the connective word “and” is called conjunction of p and q, written as p∧q. • The conjunction of p and q is true if both p and q are true otherwise it is false. p q p∧q 1 1 1 1 0 0 0 1 0 0 0 0
- 10. Disjunction ∨ • A compound proposition of p and q formed using the connective word “or” is called disjunction of p and q, written as p∨q. • The disjunction of p and q is false if both p and q are false otherwise it is true. p q p∨q 1 1 1 1 0 1 0 1 1 0 0 0
- 11. Common properties of ∧ and ∨
- 12. Common properties of ∧ and ∨
- 13. Common properties of ∧ and ∨