Discrete Mathematics Tutorials

What is Discrete Mathematics?

Discrete mathematics is the study of mathematical structures that are fundamentally discrete in nature. Mathematical structures can be discrete and continuous, and Discrete mathematics concerns with mathematical structures that are discrete.

discrete math tutorials

LOGIC:

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

Proposition:

A statement is a declarative sentence that makes a statement true or false and this statement is knowns as a proposition.

If a proposition is true, then the truth value of the proposition will be “true”.

If a proposition is false, then the truth value of the proposition will be “false”.

The truth values “true” can be denoted by the letter T.

The truth values “false”  can be denoted by the letter F.

EXAMPLES:

Propositions

  • The milk color is white. (T)
  • 3+4=7 (T)
  • 8+2=11 (F)
  • There are four fingers in a hand. (F)

Not Propositions

  • Open the eyes.
  • He is a naughty boy.
  • She is a very decent girl.

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 = 5, x > 9, Here, “x > 9” is a statement with truth-value FALSE.

What are STATEMENTS and what are not STATEMENTS?

1) x + 8 is positive. Not a statement
2) May I visit your hostel? Not a statement
3) The movie is funny. A statement
4) It is Cool today. A statement
5) 5>9 A statement
6) x=3, y=5, x + y = 12 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 for me”

Note: AND, OR, NOT are called LOGICAL CONNECTIVES.

SYMBOLIC REPRESENTATION

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

EXAMPLES:

p = “The UK stands for the United Kingdom”

q = “2 + 2 = 4”

CONNECTIVE MEANINGS SYMBOLS CALLED
Negation not ~ Tilde
Conjunction and Hat
Disjunction or Vel
Conditional if…then… Arrow
Biconditional if and only if Double arrow

EXAMPLES of LOGICAL CONNECTIVES

p = “The UK stands for the United Kingdom”

q = “2 + 2 = 4”

p q = “UK stands for United Kingdom and 2 + 2 = 4

p q = “UK stands for United Kingdom or 2 + 2 = 4

~p = “It is not the case that the UK stands for the United Kingdom”

~ q = “It is not the case that 2 + 2 = 4”

TRANSLATING FROM ENGLISH TO SYMBOLS

Let p = “It is cool”, and q = “ It is rainy”
SENTENCE SYMBOLIC FORM
1. It is not cool. ~ p
2. It is cool and rainy. p q
3. It is cool or rainy. p q
4. It is not cool but rainy. ~ p q
5. It is neither cool nor rainy. ~ p ~ q

EXAMPLE

Let        X = “Her big son is healthy”

Y = “Her big son is wealthy”

Z = “Her big son is wise”

Translate the compound statements to symbolic form:

  • Her big son is healthy and wealthy but not wise.
    • (X∧ Y) ∧ (~ Z)
  • Her big son is not wealthy but he is healthy and wise.
    • Y ∧ (X ∧ Z)
  • Her big son is neither healthy, wealthy nor wise.
    • X ∧ ~ Y ∧ ~ Z

TRANSLATING FROM SYMBOLS TO ENGLISH

Let      m = “Ali is good in C++”

c = “Ali is a Software Engineering student”

Translate the following statement forms into plain English:

1) ~ c Ali is not a Software Engineering student
2) c∨ m Ali is a Software Engineering student or good in Mathematics.
3) m ∧ ~ c Ali is good in C++” but not a Software Engineering student

We can analyze a compound statement by making a truth table for it.

NEGATION (~):

If p is a statement variable, then the negation of  p means“not p” and negation of p can be denoted as “~p”

It makes the True as False and converts the False to True.

TRUTH TABLE FOR ~ p

truth table negation P and Q

CONJUNCTION ():

If p and q are statements, then the conjunction of p and q can be denoted as “p q”.

Remarks

  • p q is true only when both p and q are true.
  • If either p or q is false, or both are false, then p q is false.

Truth Table AND p ∧ q

INCLUSIVE OR DISJUNCTION ()

If p & q are statements, then the disjunction of p and q can be denoted as “p q”.

Remarks:

  • p q is true when at least one of p or q is true.
  • p q is false only when both p and q are false.

TRUTH TABLE FOR DISJUNCTION (∨) or INCLUSIVE OR

Truth Tables examples

Truth Tables examples

TRUTH TABLE FOR EXCLUSIVE OR examples

The statement “p or q” means “p or q but not both” or “p or q and not p and q” which translates into symbols as (p q) ~ (p q).  It can be represented as p q or p XOR q.

TRUTH TABLE FOR EXCLUSIVE OR examples

Truth Tables excercise solution

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