logic, preposition etc

some topic of discrete structure

1. Discrete Structures By Abdur rehman usmani

2. Logic Study of the principles and techniques of reasoning Basis of all mathematical reasoning, and of all automated reasoning. Foundation for computer science operation. For reasoning about their truth or falsity.

3. Proposition A proposition is a *declarative sentence/statement that is either true or false, but not both. Islamabad is the capital of the Pakistan. 1 + 1 = 2 If 1=2 then roses are red. These are not proposition What time is it ? x + 1 = 2

4. Proposition We use letters to denote propositional variables (or statement variables ), that is p , q, r, s, . . . . We say that the truth value of a proposition is either true ( T ) or false ( F ).

5. Proposition “Today is January 27” Is this a statement? Yes Is this a proposition? yes What is the truth value of the proposition? false

6. Connectives/Logical Operators Compound propositions(statements), are formed from existing propositions using logical operators .

7. Truth Table The truth value of the compound proposition depends only on the truth value of the component propositions. Such a list is a called a truth table.

8. Negation (NOT )  If p = “I have brown hair .” then ¬p = “I do not have brown hair.” P  P true (T) false (F) false (F) true (T)

9. Conjunction (AND)  If p=“I will have salad for lunch.” and q=“I will have biryani for dinner .”, then p∧q =“I will have salad for lunch and AND I will have biryani for dinner.” P Q P  Q T T T T F F F T F F F F

10. Disjunction (OR)  p=“My car has a bad engine.” q=“My car has a bad carburetor.” p∨q =“Either my car has a bad engine, or my car has a bad carburetor.” P Q P  Q T T T T F T F T T F F F

11. Connectives Let p= “It rained last night”, q =“The sprinklers came on last night,” r =“The lawn was wet this morning.” Translate each of the following into English: ¬p = “It didn’t rain last night.” r ∧ ¬p = “The lawn was wet this morning, and it didn’t rain last night.” ¬ r ∨ p ∨ q = “Either the lawn wasn’t wet this morning, or it rained last night, or the sprinklers came on last night .”

12. Connectives Let p = “It is hot” q =“ “It is sunny” It is not hot but it is sunny. It is neither hot nor sunny . Solution ⌐ p∧q ⌐ p∧ ⌐q

13. Exclusive Or (XOR )  p = “I will earn an A in this course,” q = “I will drop this course,” p ⊕ q = “I will either earn an A in this course, or I will drop it (but not both!)” P Q P  Q T T F T F T F T T F F F

14. Exclusive Or (XOR )  The exclusive or of p and q , denoted by p ⊕ q , is the proposition that is true when exactly one of p and q is true and is false otherwise. P Q P  Q T T F T F T F T T F F F p = “I will earn an A in this course,” q = “I will drop this course,” p ⊕ q = “I will either earn an A in this course, or I will drop it (but not both!)”

15. Implication (if - then )  The conditional statement p → q is the proposition “if p, then q .” The conditional statement p → q is false when p is true and q is false, and true otherwise. p is called the hypothesis and q is called the conclusion

16. Implication (if - then )  P Q P  Q T T T T F F F T T F F T p = “You study hard.” q = “You will get a good grade.” p → q = “If you study hard, then you will get a good grade .”

17. Biconditionals (if and Only If)  p = “ Zardari wins the 2008 election.” q = “ Zardari will be president for five years.” p ↔ q = “If, and only if, Zardari wins the 2008 election, Zardari will be president for five years .” p ↔ q does not imply that p and q are true, or that either of them causes the other, or that they have a common cause.