# Preposition logic | Discrete Mathematics

In this article, we will learn about the prepositions and statements and some basic logical operation in discrete mathematics.
Submitted by Prerana Jain, on August 31, 2018

## Preposition or Statement

A preposition is a definition sentence which is true or false but not both.

For example: The following 8 sentences,

1. Paris in France
2. 2 + 2 =4
3. London in Denmark
4. X = 2 is solution of x^2 = 4
5. 1 + 1 = 2
6. 9<6
7. Where are you going?

All of them are preposition except vii and viii moreover i, ii and vi are true whereas iii, iv, v are false.

### Compound proposition

Many propositions are composite that is composed of subpropositions and various connectives discussed subsequently. Such a composite proposition is said to be compound propositions. A proposition is called primitive if it cannot be broken down into the simpler proposition that is if it is not composite.

Example:

1. "John intelligent or studies every night" is a compound proposition with subproposition. "John is intelligent" and "john studies every night".
2. "Roses are red and violets are blue" is a compound proposition with subproposition "Roses are red" and "violets are blue".

### Basic logical operation

The Three basic logical operations conjunction, disjunction, and negation which corresponds respectively. To the English words "and", "or" and "not".

1) Conjunction (p ^ q):

Any two proposition can be combined by the word and to form a compound proposition said to be the conjunction of the original proposition. Symbolically p ^ q read p and q denotes the conjunction of p and q. Since, p ^ q is a proposition it has the truth value and this truth value depends only on the truth values of p and q, specifically:

Definition: If p and q are true then p ^ q is true otherwise p ^ q is false.

pqp ^ q
TTT
TFF
FTF
FFF

Example: Consider the following 4 statements:

1. Paris is in France and 2+2 = 4
2. Paris is in France and 2 + 2 = 5
3. Paris is in England and 2 + 2 = 4
4. Paris is in England and 2 + 2 = 5

In the given four statements only the first statement is true. Each of the other statements is false since at least one of its substatements is false.

2) Disjunction (p V q)

Any two proposition can be combined by the word "or" to form a compound proposition is said to be the disjunction of the original proposition, symbolically p V q.

Read "p or q" denotes the disjunction of p and q. The truth value of p V q depends only on the truth values of p and q as follow:

Definition: If p and q are false then p V q is false, otherwise p V q is true.

pqpVq
TTT
TFT
FTT
FFF

Example: Consider the following four statements:

1. Paris is in France or 2 + 2 = 4
2. Paris is in France or 2 + 2 = 5
3. Paris is in England or 2 + 2= 4
4. Paris is in England or 2 + 2 = 5

Only the last statements are false. Each of the other statements is true since at least of its substatements is true.

3) Negation( ~p)

Given any proposition p another proposition is said to be the negation of p can be formed by writing - it is not the case that... or "it is false that ...", before p or if possible by inserting in p the word "not" symbolically. ~p or ~p.

Read "not p", denotes the negation of p. The truth value of p depends on the truth value of p as follows:

Definition: If p is true then ~p is false and if p is false then ~p is true.

p ~p
T F
F ~F