Upcoming SlideShare
×

# Disrete mathematics and_its application_by_rosen _7th edition_lecture_1

474

Published on

Published in: News & Politics, Technology
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total Views
474
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
25
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Disrete mathematics and_its application_by_rosen _7th edition_lecture_1

1. 1. LECTURE 1
2. 2. Disrete mathematics and its application by rosen 7th edition THE FOUNDATIONS: LOGIC AND PROOFS 1.1 PROPOSITIONAL LOGIC
3. 3.  A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both  1 + 1 = 2 (true)  4 + 9 = 13 (true)  Islamabad is capital of Pakistan (true)  Karachi is the largest city of Pakistan (true)  100+9 = 111 (false)  Some sentences are not prepositions  Where is my class? (un decelerated sentence)  What is the time by your watch? (un decelerated sentence)  x + y = ? ( will be prepositions when value is assigned)  Z +w * r = p PROPOSITIONS
4. 4.  We use letters to denote propositional variables (or statement variables).  The truth value of a proposition is true, denoted by T, if it is a true proposition.  The truth value of a proposition is false, denoted by F, if it is a false proposition.  Many mathematical statements are constructed by combining one or more propositions. They are called compound propositions, are formed from existing propositions using logical operators. PROPOSITIONS
5. 5.  Definition: Let p be a proposition. The negation of p, denoted by￢p (also denoted by p), is the statement “It is not the case that p.”  The proposition ￢p is read “not p.” The truth value of the negation of p, ￢p, is the opposite of the truth value of p.  Also denoted as “ ′ ”  Examples:  p := Sir PC is running Windows OS  ￢p := sir PC is not running Windows OS  p := a + b = c  p := a + b ≠ c NEGATION The Truth Table for the Negation of a Proposition p ￢p T F F T
6. 6.  Definition: Let p and q be propositions. The conjunction of p and q, denoted by p ∧ q, is the proposition “p and q.” The conjunction p ∧ q is true when both p and q are true and is false otherwise.  Also known as UNION, AND, BIT WISE AND, AGREGATION  Denoted as ^ , &, AND CONJUNCTION The Truth Table for the conjunction of a Proposition p q p ^ q T T T F F T F F T F F F
7. 7.  Definition: Let p and q be propositions. The disjunction of p and q, denoted by p ∨ q, is the proposition “p or q.” The disjunction p ∨ q is false when both p and q are false and is true otherwise.  Also known as OR, BIT WISE OR, SEGREGATION  Denoted as v , || , OR DISJUNCTION The Truth Table for the conjunction of a Proposition p q p v q T T T F F T F F T T T F
8. 8.  Definition: Let p and q be propositions. 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.  Also known as ZORING  Denoted as XOR , Ex OR, ⊕ EXCLUSIVE OR The Truth Table for the conjunction of a Proposition p q p ⊕ q T T T F F T F F F T T F
9. 9.  Let p and q be propositions. 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.  In the conditional statement p → q, p is called the hypothesis (or antecedent or premise) and q is called the conclusion (or consequence).  Denoted by  CONDITIONAL STATEMENT The Truth Table for the conjunction of a Proposition p q p → q T T T F F T F F T F T T
10. 10.  The proposition q → p is called the converse of p → q.  The converse, q → p, has no same truth value as p → q for all cases.  Formed from conditional statement. CONVERSE
11. 11.  The contrapositive of p → q is the proposition ￢q →￢p.  only the contrapositive always has the same truth value as p → q.  The contrapositive is false only when ￢p is false and ￢q is true.  Formed from conditional statement. CONTRAPOSITIVE The Truth Table for the CONTRAPOSITIVE of a Proposition p q ￢p ￢q ￢p → ￢q T T F F T F F T F T T F F F T T T T F T
12. 12.  Formed from conditional statement.  The proposition ￢p →￢q is called the inverse of p → q.  The converse, q → p, has no same truth value as p → q for all cases. INVERSE
13. 13.  Let p and q be propositions. The biconditional statement p ↔ q is the proposition “p if and only if q.” The biconditional statement p ↔ q is true when p and q have the same truth values, and is false otherwise.  Biconditional statements are also called bi-implications. BICONDITIONAL The Truth Table for the CONTRAPOSITIVE of a Proposition q p q ↔ p T T T F F T F F T F F T
14. 14.  Definition: When more that one above defined preposition logic combines it is called as compound preposition.  Example:  (p^q)v(p’)  (p ⊕ q) ^ (r v s) COMPOUND PROPOSITIONS
15. 15.  (p ∨￢q) → (p ∧ q) COMPOUND PROPOSITIONS (TRUTH TABLE) The Truth Table of (p ∨￢q) → (p ∧ q) p q ￢q p ∨￢q p ∧ q (p ∨￢q) → (p ∧ q) T T F F T F T F F T F T T T F T T F F F T F T F
16. 16. Precedence of Logical Operators. Operator Precedence ￢ 1 ^ 2 v 3 → 4 ↔ 5 XOR 6 PRECEDENCE OF LOGICAL OPERATORS
17. 17.  Computers represent information using bits  A bit is a symbol with two possible values, namely, 0 (zero) and 1 (one).  A bit can be used to represent a truth value, because there are two truth values, namely, true and false.  1 bit to represent true and a 0 bit to represent false. That is, 1 represents T (true), 0 represents F (false).  A variable is called a Boolean variable if its value is either true or false. Consequently, a Boolean variable can be represented using a bit. LOGIC AND BIT OPERATIONS Truth Value Bit T 1 F 0
18. 18.  Computer bit operations correspond to the logical connectives.  By replacing true by a one and false by a zero in the truth tables for the operators ∧ (AND) , ∨ (OR) , and ⊕ (XOR) , the tables shown for the corresponding bit operations are obtained. LOGIC AND BIT OPERATIONS Table for the Bit Operators OR, AND, and XOR. p q p ^ q p v q p XOR q 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 1 0
19. 19.  01 1011 0110 11 0001 1101 11 1011 1111 bitwise OR 01 0001 0100 bitwise AND 10 1010 1011 bitwise XOR  11 1010 1110 11 0001 1101 11 1011 1111 bitwise OR 11 0000 1100 bitwise AND 00 1010 0011 bitwise XOR BITWISE OR, BITWISE AND, AND BITWISE XOR
1. #### A particular slide catching your eye?

Clipping is a handy way to collect important slides you want to go back to later.