# 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.

__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:**

- “3 + 2 = 5”
**and**“Lahore is a city in Pakistan” - “The grass is green”
**or**“ It is hot today” - “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**

__CONJUNCTION (____∧____):__

If ** p** and

**are statements, then the conjunction of**

*q***and**

*p***can be denoted as**

*q*

*“p***∧**

*q”.***Remarks**

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

__INCLUSIVE OR DISJUNCTION (____∨____)__

If ** p** &

**are statements, then the disjunction of**

*q***and**

*p***can be denoted as**

*q*

*“p***∨**

*q”.***Remarks:**

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

## 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.**

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