Computing Mathematics

Computing Mathematics

Agenda • Your Lecturer and tutors • Module Objective • Module Assessments and Syllabus Summary • Recommended readings

Your Lecturer and Tutor Are… Mr.Ashok Dhungana Mr. Roshan Chaudhary (Islington College) Tutor (Islington College) Sr. Lecturer

Me And Islington…

Syllabus • Week 1: Introduction to logic • Week 2 :Sets and Probability • Week 3 :Number Bases • Week 4 : Relations and Functions • Week 5 : Methods of proofs Each week there will be one lecture and one tutorial.

Module Assessments • Assessments: 100% EXAM (Week 6 ) Its a closed book unseen exam of 2 hours . Note:- Students should obtain 40% on examination to pass.

INTRODUCTION TO LOGIC

Introduction to Logic • what logic is; • about the basis of logic: the simple statement; • how to construct a truth table; • about the five basic logic connectives ; • what makes a statement a tautology; • what makes a statement a contradiction; • what logical equivalence is; • what makes an argument valid;

WHAT IS LOGIC? • Logic is the study of the principles and methods used in distinguishing valid arguments from those that are invalid. • Logic is also known as propositional calculus.

SIMPLE STATEMENTS • The basic building block in logic is the statement, also referred to as a proposition. • A statement is a declarative sentence, which can only be either true or false. • Statements are represented by letters such as p, q, and r, . . . • Example: Jakarta is a city in Indonesia

COMPOUND STATEMENTS • The combination of two or more simple statements is a compound statement, or compound proposition. • Example: “2 + 1 = 5” and “6 + 2 = 8” • Example: “The sky is clear” or “It is raining today” • The variables p, q, r, . . . denote simple statements in the compound proposition , where P is a proposition.

BASIC LOGIC CONNECTIVES Compound statements are connected using mainly five basic connectives: • conjunction • disjunction • negation • conditional • bi conditional

CONJUNCTION • Any two statements can be combined by the word “and” to form a composite statement which is called the conjunction of the original statements. • The connection of the statements p and q is symbolically represented by pΛq • If p is true and q is true then pΛq is true; otherwise pΛq is false

CONJUNCTION Example 1: Sidney is in Australia and 2 + 2 = 4 Example 2: Sidney is in Australia and 2 + 2 = 5 Example 3: Sidney is in Malaysia and 2 + 2 = 4 Example 4: Sidney is in Malaysia and 2 + 2 = 5 • Only example 1 contains two simple statements which are true, the others all contain simple statements in which at least one of them is false, so only example 1 is true.

DISJUNCTION • Any two statements can be combined by the word “or” , to form a new statement which is called the disjunctionof the original two statements. • The connection of the statements p or q is symbolically represented by pνq • If p is true or q is true or both p and q are true, then pνq is true; otherwise pνq is false.

DISJUNCTION Example 1: Sidney is in Australia or 2 + 2 = 4 Example 2: Sidney is in Australia or 2 + 2 = 5 Example 3: Sidney is in Malaysia or 2 + 2 = 4 Example 4: Sidney is in Malaysia or 2 + 2 = 5 • Only example 4 is false. Each of the other compound statements is true since at least one of its simple statements is true.

NEGATION • Given any statement p, another statement, called the negation of p, can be formed by writing “It is false that . . .” before p or, if possible, by inserting in p the word “not” • Negation can be symbolically represented by ~p, or ┐p • The truth values of ~p in a truth table: • If p is true then ~p is false; if p is false, then ~p is true. Example : If p is ‘Sidney is in Australia’, then ‘Sidney is not in Australia’ is the negation ~p

CONDITIONAL Many statements, especially in mathematics, are of the form “If p then q” or “p implies q”. Such statements are calledconditional statements. Conditional statements are symbolically represented as p→q The conditional p→q is true unless p is true and q is false.

Conditional Example 1: If Sidney is in Australia then 2 + 2 = 4 Example 2: If Sidney is in Australia then 2 + 2 = 5 Example 3: If Sidney is in Malaysia then 2 + 2 = 4 Example 4: If Sidney is in Malaysia then 2 + 2 = 5 • By the conditional p → q only example 2 is false. But how can this be as clearly ‘2 +2 = 4’ is true and ‘2 + 2 = 5’ is clearly false?

BICONDITIONAL • Another common statement called a bi conditional statementis of the form “p iff q” • Bi conditional statements are symbolically represented as p↔q • If p and q have the same truth value, then p↔q is true; • if p and q have opposite truth values, then p↔q is false.

BICONDITIONAL Example 1: Sidney is in Australia iff 2 + 2 = 4 Example 2: Sidney is in Australia iff 2 + 2 = 5 Example 3: Sidney is in Malaysia iff 2 + 2 = 4 Example 4: Sidney is in Malaysia iff 2 + 2 = 5 By the conditional p ↔ q examples 1 & 4 are true, examples 2 & 3 are false. pq can be represented as

BASIC LOGIC CONNECTIVES

PROPOSITIONS AND TRUTH TABLES Example : The truth table of the proposition is: Observe that the first columns of the table are for the variables p, q, . . . and there are enough rows in the table to allow for all possible combinations of T and F for these variables, i.e. the number of rows = Construct a truth table for the following: 1. (a ᴧ ~b) ↔ (a ᴧ b) 2. (a v b) → ~c

TAUTOLOGY • A compound proposition that is always TRUE is called a tautology. • Example : p Ú ~p is a tautology as all entries in the last column are T’s Show that the following are tautologies : 1.(p ᴧ q) → (p v ~q) 2. [(p ↔ q) ᴧ ~p] → ~q 3. (p v ~q) ↔ ~(~p ᴧ q)

CONTRADICTION • A compound proposition that is always FALSE is called a contradiction. • Example : p Ù ~p is a contradiction as all entries in the last column are F’s Show that the following are contradictions: 1.(p → q) ᴧ ( ~q ᴧ p) 2. (p ᴧ q) ᴧ ~ (p v q) 3.(p → q) ᴧ (~q ᴧ p)

LOGICAL EQUIVALENCE • Two propositions and are said to be logically equivalent if the final columns in their truth tables are the same. • Logical equivalence is denoted with ≡ Example: Show that p → q and ~q → ~p are logically equivalent.

LOGICAL EQUIVALENCE • Example: Show that (p→q) Λ (q→p) p↔q • Since the last two columns in the truth tables are the same the statements are logically equivalent.

De MORGAN’S LAWS • De Morgan’s Laws states that: • ~ ( p ᴧ q) ≡ ~ p v ~ q • ~ ( p v q) ≡ ~ p ᴧ ~ q • Prove the De Morgan's laws using truth table.

ARGUMENTS • An argument is a relationship between a set of propositions, , called premises, and another proposition Q, called the conclusion.An argument is denoted by ├ Q • An argument ├ Q is said to be valid iff is a tautology.

Questions Determine the validity of the following arguments: 1. p v q , ~p Ⱶ q 2. p → q , q → r , ~r Ⱶ ~p 3. If you do not study you will fail your examination. You failed therefore you did not study. 4. If I am not in Malaysia, then I am not happy; if I am happy, then I am singing; I am into singing; therefore, I am not in Malaysia

Logic Gates • NOT gate • AND gate • OR gate • NAND gate • NOR gate

P P NOT P NOT P T 1 F 0 F 0 T 1 NOT P OUTPUT INPUT NOT-GATE Truth Table

p q p v q 0 0 0 0 1 1 1 0 1 1 1 1 OUTPUT p v q p INPUTS q OR-GATE Truth Table OR-GATE

0 1 0 p 1 q 1 1 0 0 0 0 p ᴧ q 0 1 INPUTS OUTPUT p ᴧ q p q AND-GATE Truth Table AND-GATE

1 A 0 1 0 0 1 0 B 1 1 1 1 0 A NAND B A NAND B p q p Symbol q A NAND B NAND-GATE Truth Table

Truth Table 1 1 0 A 0 1 0 1 0 B A NOR B 0 0 1 0 A NOR B A B A B A NOR B NOR-GATE

Summary Propositions Basic logical connectives Tautology Contradiction Arguments Logic Gates

The End

Computing Mathematics