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# Lecture 1 - PowerPoint PPT Presentation

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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The Foundations:Logic and Proofs

Disrete mathematics and its application by rosen

7th edition

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 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

• 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.

• 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

• 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

• 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

• 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, ⊕

• 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 

• 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.

• 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.

• 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.

• 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.

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)

Compound Propositions (truth Table)

• (p ∨￢q) → (p ∧ q)

• 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.

• 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.

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