25
Mon, Jun

Discrete Structure | Logic | Lecture 1

Discrete Structure
Typography

Course Objective:

  1. Express statements with the precision of formal logic.
  2. Analyze arguments to test their validity.
  3. Apply the basic properties and operations related to sets.
  4. Apply to sets the basic properties and operations related to relations and function.
  5. Define terms recursively.
  6. Prove a formula using mathematical induction.
  7. Prove statements using direct and indirect methods.
  8. Compute probability of simple and conditional events.
  9. Identify and use the formulas of combinatorics in different problems.
  10. Illustrate the basic definitions of graph theory and properties of graphs.
  11. Relate each major topic in Discrete Mathematics to an application area in computing.

Recommended Books:

  1. Discrete Mathematics with Applications (second edition) by Susanna S. Epp.
  2. Discrete Mathematics and Its Applications (fourth edition) by Kenneth H. Rosen.
  3. Discrete Mathematics by Ross and Wright.

Main Topics:

  1. Logic.
  2. Sets & Operations on sets.
  3. Relations & Their Properties.
  4. Functions.
  5. Sequences & Series.
  6. Recurrence Relations.
  7. Mathematical Induction.
  8. Loop Invariants.
  9. Combinatorics.
  10. Probability.
  11. Graphs and Trees.

Discrete

Set of Integers:

• • • • • •

3 -2 -1 0 1 2

Set of Real Numbers:

• • • • • •

-3 -2 -1 0 1 2

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.-- -- -- -- -- Not Propositions
  2. 4 + 2 = 6 -- -- -- -- -- -- -- - Close the door.
  3. 4 + 2 = 7 -- -- -- -- -- -- -- - x is greater than 2.
  4. There are four fingers
    in a hand. are propositions. -- -- -- -- -- He is very rich are not propositions.

Rule:

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

Example:

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

Understanding Statements:

  1. x + 2 is positive. -- -- -- -- -- -- -- Not a statement.
  2. 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.

Examples:-- -- -- -- --- -- -- - Logical Connectives.

  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”.

Symbolic Representation

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.
  6. Examples:

    Let h = “Zia is healthy”
    w = “Zia is wealthy”
    s = “Zia is wise”.

    Translate the compound statements to symbolic form:

    1. Zia is healthy and wealthy but not wise. -- -- -- -- -- -- -- -- -- (h /\ w) /\ (~s).
    2. Zia is not wealthy but he is healthy & wise. -- -- -- -- -- -- -- ~w /\ (h /\ s).
    3. Zia is neither healthy, wealthy nor wise. -- -- -- -- -- -- -- -- -- ~h /\ ~w /\ ~s.

    Translating From Symbols To English:

    Let m = “Ali is good in Mathematics”.

    c = “Ali is a Computer Science student”.

    Translate the following statement forms into plain English:

    1. ~ c -- -- -- -- -- - Ali is not a Computer Science student.
    2. c \/ m -- -- -- Ali is a Computer Science student or good in Maths.
    3. m /\ ~c-- -- -- - Ali is good in Maths but not a Computer Science student.

    A convenient method for analyzing a compound statement is to make a truth table for it.

    A truth table specifies the truth value of a compound proposition for all possible truth values of its constituent propositions.

    Negation (~):

    If p is a statement variable, then negation of p, “not p”, is denoted as “~p”.

    It has opposite truth value from p i.e., if p is true, ~p is false; if p is false, ~p is true.

    Truth Table For ~ p

    Discrete

    Conjunction(/\):

    If p and q are statements, then the conjunction of p and q is “p and q”, denoted as “p /\ q”.

    It is true when, and only when, both p and q are true. If either p or q is false, or if both are false, p /\ q is false.

    Truth Table For p /\ q

    Discrete

    Disjunction ( \/ ) or INCLUSIVE OR

    If p & q are statements, then the disjunction of p and q is “p or q”, denoted as “p \/ q”. It is true when at least one of p or q is true and is false only when both p and q are false.

    Truth Table For p \/ q

    Note it that in the table F is only in that row where both p and q have F and all other values are T. Thus for finding out the truth values for the disjunction of two statements we will only first search out where the both statements are false and write down the F in the corresponding row in the column of p Ú q and in all other rows we will write T in the column of p Ú q.

    Remark:

    Note that for Conjunction of two statements we find the T in both the statements, but in disjunction we find F in both the statements. In other words we will fill T first in the column of conjunction and F in the column of disjunction.

    Summary

    1. What is a statement?
    2. How a compound statement is formed.
    3. Logical connectives (negation, conjunction, disjunction).
    4. How to construct a truth table for a statement form.