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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
The Foundations:Logic and Proofs

Disrete mathematics and its application by rosen

7th edition

1.1 Propositional Logic


Propositions
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


Propositions1
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.


Negation
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


Conjunction
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


Disjunction
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


Exclusive or
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, ⊕


Conditional statement
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 


Converse
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.


Contrapositive
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.


Inverse
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.


Biconditional
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.


Compound propositions
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
Compound Propositions (truth Table)

  • (p ∨¬q) → (p ∧ q)



Logic and bit operations
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.


Logic and bit operations1
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.


Bitwise or bitwise and and bitwise xor
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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