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Lecture 1

Lecture 1 . The Foundations: Logic and Proofs. Disrete mathematics and its application by rosen 7 th edition. 1.1 Propositional Logic. Propositions. A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both

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Lecture 1

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  1. Lecture 1

  2. The Foundations:Logic and Proofs Disrete mathematics and its application by rosen 7th edition 1.1 Propositional Logic

  3. Propositions • 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? (undeceleratedsentence) • What is the time by your watch? (undeceleratedsentence) • x + y = ? ( willbeprepositionswhenvalueisassigned) • Z +w * r = p

  4. Propositions • 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.

  5. negation • 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

  6. conjunction • 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

  7. disjunction • 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

  8. exclusive or • 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 asZORING • Denoted as XOR , Ex OR, ⊕

  9. conditional statement • 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 

  10. CONVERSE • 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.

  11. CONTRAPOSITIVE • 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.

  12. INVERSE • 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.

  13. biconditional • 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.

  14. Compound Propositions • 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)

  15. Compound Propositions (truth Table) • (p ∨￢q) → (p ∧ q)

  16. Precedence of Logical Operators

  17. Logic and Bit Operations • 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.

  18. Logic and Bit Operations • 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.

  19. bitwise OR, bitwise AND, and bitwise XOR • 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

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