MTH001 Official Handouts

TABLE OF CONTENTS

Lecture 1: Introduction ----------------------------------------------------------------------------- Lecture 2: Truth Tables ---------------------------------------------------------------------------- Lecture 3: Laws of Logic -------------------------------------------------------------------------- Lecture 4: Biconditional operator ---------------------------------------------------------------- Lecture 5: Arguments ------------------------------------------------------------------------------ Lecture 6: Sets --------------------------------------------------------------------------------------- Lecture 7: Sets Continued ------------------------------------------------------------------------ Lecture 8: Sets Continued ------------------------------------------------------------------------ Lecture 9: Sets Continued ------------------------------------------------------------------------ Lecture 10: Sets Continued ---------------------------------------------------------------------- Lecture 11: Relations and Functions ----------------------------------------------------------- Lecture 12: Functions ------------------------------------------------------------------------------ Lecture 13: Sequences ---------------------------------------------------------------------------- Lecture 14: Series ----------------------------------------------------------------------------------- Lecture 15: Applications of Basic Mathematics ---------------------------------------------- Lecture 16: Applications of Basic Mathematics ---------------------------------------------- Lecture 17: Applications of Basic Mathematics ---------------------------------------------- Lecture 18: Applications of Basic Mathematics ---------------------------------------------- Lecture 19: Matrices -------------------------------------------------------------------------------- Lecture 20: Operation on Matrices -------------------------------------------------------------- Lecture 21: Ratios and Proportions ------------------------------------------------------------- Lecture 22: Introduction to Statistics ----------------------------------------------------------- Lecture 23: Data Representation ---------------------------------------------------------------- Lecture 24: The Central Tendency of a data set -------------------------------------------- Lecture 25: Median, Empirical Relation, Quartiles ------------------------------------------ Lecture 26: Geometric mean, Harmonic mean & relationship between them ------- Lecture 27: Measures of Dispersion ------------------------------------------------------------ Lecture 28: Mean Deviation, Standard Deviation and Variance & Coefficient of variation ------------------------------------------------------------- Lecture 29: Permutations, Combinations ----------------------------------------------------- Lecture 30: Definitions of Probability ----------------------------------------------------------- Lecture 31: Relative Frequency, Axiomatic Definition & Laws of Probability --------- Lecture 32: Independent and Dependent Events, Multiplication Theorem & Marginal Probability ------------------------------

LECTURE # 1 · Introduction · 1.Recommended Books:

1. Discrete Mathematics with Applications (second edition) by Susanna S. Epp

MAIN TOPICS:

1. Logic
2. Sets & Operations on sets
3. Relations & Their Properties
4. Functions
5. Sequences & Series

Set of Integers:

-2 -1

Set of Real Numbers:

-3 -2 -1

What is Discrete Mathematics?:

Discrete Mathematics concerns processes that consist of a sequence of individual steps.

LOGIC:

Logic is the study of the principles and methods that distinguishes between a valid and an invalid argument.

SIMPLE STATEMENT:

A statement is a declarative sentence that is either true or false but not both. A statement is also referred to as a proposition Example: 2+2 = 4, It is Sunday today If a proposition is true, we say that it has a truth value of "true”. If a proposition is false, its truth value is "false". The truth values “true” and “false” are, respectively, denoted by the letters T and F.

EXAMPLES:

1. Grass is green.
2. 4 + 2 = 6
3. 4 + 2 = 7
4. There are four fingers in a hand.

are propositions

Rule:

If the sentence is preceded by other sentences that make the pronoun or variable reference clear, then the sentence is a statement.

Not Propisitions

Close the door. x is greater than 2. He is very rich are not propositions.

Example

Bill Gates is an American He is very rich He is very rich is a statement with truth-value TRUE.

Example:

x = 1 x > 2 x > 2 is a statement with truth-value FALSE.

UNDERSTANDING STATEMENTS:

1. x + 2 is positive. Not a statement

Q2. May I come in?

Not a statement

3. Logic is interesting. A statement
4. It is hot today. A statement
5. -1 > 0 A statement
6. x + y = 12 Not a statement

COMPOUND STATEMENT:

Simple statements could be used to build a compound statement.

LOGICAL CONNECTIVES:

EXAMPLES

1. “3 + 2 = 5” and “Lahore is a city in Pakistan”
2. “The grass is green” or “ It is hot today”
3. “Discrete Mathematics is not difficult to me”

AND, OR, NOT are called LOGICAL CONNECTIVES.

SYMBOLIC REPRESENTATION:

Statements are symbolically represented by letters such as p, q, r,...

EXAMPLES:

p = “Islamabad is the capital of Pakistan” q = “17 is divisible by 3”

Double arrow · if and only if · Biconditional · Arrow · if…then… · Conditional · Vel · or · Hat · and · Tilde · not · Negation · CALLED · SYMBOL · MEANINGS · CONNECTIVE · EXAMPLES:

p = “Islamabad is the capital of Pakistan” q = “17 is divisible by 3” p ∧ q = “Islamabad is the capital of Pakistan and 17 is divisible by 3” p ∨ q = “Islamabad is the capital of Pakistan or 17 is divisible by 3” ~p = “It is not the case that Islamabad is the capital of Pakistan” or simply “Islamabad is not the capital of Pakistan”

TRANSLATING FROM ENGLISH TO SYMBOLS:

Let p = “It is hot”, and q = “ It is sunny”

SENTENCE

SYMBOLIC FORM

1. It is not hot. ~ p
2. It is hot and sunny. p ∧q
3. It is hot or sunny. p ∨ q
4. It is not hot but sunny. ~ p ∧q
5. It is neither hot nor sunny. ~ p ∧ ~ q