LECTURE 1
Disrete
mathematics
and its
application by
rosen
7th edition
THE FOUNDATIONS:
LOGIC AND PROOFS
1.1 PROPOSITIONAL
LOGIC
 A proposition is a declarative sentence (that is, a sentence
that declares a fact) that is either true or false, but not...
 We use letters to denote propositional variables (or statement
variables).
 The truth value of a proposition is true, d...
 Definition: Let p be a proposition. The negation of p, denoted
by¬p (also denoted by p), is the statement “It is not the...
 Definition: Let p and q be propositions. The conjunction of p
and q, denoted by p ∧ q, is the proposition “p and q.” The...
 Definition: Let p and q be propositions. The disjunction of p
and q, denoted by p ∨ q, is the proposition “p or q.” The
...
 Definition: Let p and q be propositions. The exclusive or of p
and q, denoted by p ⊕ q, is the proposition that is true ...
 Let p and q be propositions. The conditional statement p → q
is the proposition “if p, then q.” The conditional statemen...
 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
c...
 The contrapositive of p → q is the proposition ¬q →¬p.
 only the contrapositive always has the same truth value as p
→ ...
 Formed from conditional statement.
 The proposition ¬p →¬q is called the inverse of p → q.
 The converse, q → p, has n...
 Let p and q be propositions. The biconditional statement p ↔
q is the proposition “p if and only if q.” The biconditiona...
 Definition: When more that one above defined preposition
logic combines it is called as compound preposition.
 Example:...
 (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) → ...
Precedence of Logical Operators.
Operator Precedence
¬ 1
^ 2
v 3
→ 4
↔ 5
XOR 6
PRECEDENCE OF LOGICAL OPERATORS
 Computers represent information using bits
 A bit is a symbol with two possible values, namely, 0 (zero)
and 1 (one).
...
 Computer bit operations correspond to the logical
connectives.
 By replacing true by a one and false by a zero in the t...
 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 00...
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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
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