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# Propositions

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### Propositions

1. 1. Propositions•Is a statement or a declaration that has truth value.•It can be either True or False•But NEVER both! true (denoted either T or 1) false (denoted either F or 0).
2. 2.  Variables are used to represent propositions. The most common variables used are p, q, and r Example of Propositions:  All cows are brown.  The Earth is further from the sun than Venus.  There is life on Mars.  x + 2 = 2x when x = 2  2x2=5
3. 3. Here are some sentences that are not propositions.1. “Do you want to go to the movies?"(Since a question is not a declarative sentence, it fails to be a proposition.)2. “Clean up your room."(Likewise, an imperative is not a declarativesentence; hence, fails to be a proposition.)3. “2x = 2 + x." This is a declarative sentence,but unless x is assigned a value or is otherwiseprescribed, the sentence neither true nor false,hence, not a proposition.
4. 4. 4. “This sentence is false."What happens if you assume this statement istrue? false? This example is called a paradox andis not a proposition, because it is neither true norfalse.
5. 5. Which of the following are propositions? (a) 17 + 25 = 42 (b) July 4 occurs in the winter in the Northern Hemisphere. (c) The population of the United States is less than 250 million. (d) Is the moon round? (e) 7 is greater than 12. (f) x is greater than y.
6. 6. 1. Unary Operator Negation: “not p", ¬ p Another notation commonly used for the negation of p is ~ p2. Binary Operator
7. 7. Binary Operator: (a) conjunction: “p and q", p ∧ q. (b) disjunction: “p or q", p ∨ q. (c) exclusive or: “exactly one of p or q", p xor q", p ⊕ q. (d) implication: “if p then q", p q. (e) biconditional: “p if and only if q", p q.
8. 8.  We can create compound propositions using propositional variables, such as p; q; r; s; and connectives or logical operators. A logical operator is either a unary operator, meaning it is applied to only a single proposition; or a binary operator, meaning it is applied to two propositions. Truth tables are used to exhibit the relationship between the truth values of a compound proposition and the truth values of its component propositions.
9. 9. NEGATION: Negation OperatorExample : p: This book is interesting.¬ p can be read as: (i.) This book is not interesting. (ii.) This book is uninteresting.
10. 10. CONJUNCTION: Conjunction OperatorExample: p: This book is interesting. q: I am staying at home.p ^ q: This book is interesting, and I am staying athome.
11. 11. DISJUNCTION: Disjunction Operator Example: p: This book is interesting. q: I am staying at home.p ∨ q: This book is interesting, or I am staying athome.
12. 12. ECLUSIVE OR: Exclusive Or Operator EXAMPLE: p: This book is interesting. q: I am staying at home.p ⊕ q: Either this book is interesting, or I am stayingat home, but not both.
13. 13. Implications. Implication Operator EXAMPLE: p: This book is interesting. q: I am staying at home.p q: If this book is interesting, then I am staying athome.
14. 14. p is called the premise, hypothesis, or the antecedent.q is called the conclusion or consequent.q  p is the converse of p q.-p  -q is the inverse of p  q-q  -p is the contrapositive of p  q.
15. 15. Example: If this book is interesting, then I am staying at home. Converse: If I am staying at home, then this book is interesting. q  p is the converse of p q
16. 16. Example: If this book is interesting, then I am staying at home. Inverse: If this book is not interesting, then I am not staying at home. -p  -q is the inverse of p  q
17. 17. Example: If this book is interesting, then I am staying at home. Contrapositive: If I am not staying at home, then this book is not interesting. -q  -p is the contrapositive of p  q
18. 18. Biconditional Operator : "if and only if",has symbol p: This book is interesting. q: I am staying at home. p q: This book is interesting if and only if I am staying at home.
19. 19. The biconditional statement is equivalent to (p  q) ^ (q p)