Logic, Sets and Real Number System
1.1 Logic
The word logic is derived from the word "logos" which means reason. Man learnt to talk or give reason or argue systematically either on the basis of his experience and activities or on the basis of the idea or supposition or assumption he makes. Of these two ways the first one is known as "inductive process of reasoning" and the second one as "deductive process of reasoning".
The dictionary meaning of logic is the science of reasoning. In mathematics, we deal with various theorems and formulae. The different procedures used in giving the proofs of various theorems and formulae are based on sound reasoning. The study of such procedures based on sound reasoning is known as a logic. Logic tells us the truth and the falsity of the particular statement. Logic is the process by which we arrive at a conclusion from the given statement with a valid reason. In logic, we use symbols for words, statements and their relations to get the required result. Hence, logic is known as mathematical logic or symbolic logic.
1.2 Statements
An assertion expressed in words or symbols, which is either true or false but not both at the same time, is known as a statement.
Some examples of statements are given below:
- Water is essential for health.
- 2+4=6
- A quadrilateral has three sides.
(i), (ii) and (iii) are statements as (i) and (ii) are true but (iii) is false.
The sentences of the following type are not the statements because they do not declare the truth or falsity.
- Knock at the door.
- What is your name?
- How beautiful your country is!
In the context of logic, statements cannot be imperative, interrogative and exclamatory.
A sentence whose truth or falsity can be decided only after filling the gap in the sentence or substituting the value of the variable is known as an open sentence, otherwise it is known as closed. All the sentences considered above ((i), (ii) and (iii)) are closed and hence are statements.
The examples of open sentences are
- ... is the son of Dasharath
- x+3=5
These are not the statements.
There are two types of statements: Simple and Compound.
Simple Statement
A statement which declares only one thing is known as a simple statement. That is, a sentence that cannot be divided into two or more sentences is known as a simple statement.
- Laxmi Prasad Devkota is a great poet.
- 2×3=6
(i) and (ii) are the examples of simple statements. In mathematical logic, simple statements are denoted by the letters: p, q, r, etc.
Compound Statement
A combination of two or more simple statements is known as a compound statement. Each simple statement is known as a component of the compound statement.
The examples of compound statements are as follows.
- Nepal is in Asia and Mt. Everest is the highest peak in the world.
- 3-2=1 and 5>6
Compound statements are constructed from simple statements by means of logical connectives.
1.3 Truth value and truth table
A truth or the falsity of a statement is known as its truth value. T or F is the truth value of a statement according as it is true or false. The truth value of a simple statement depends upon the truth or falsity of the given statement. But in a compound statement, its truth value depends not only on the truth or falsity of the component statements but also on the connectives with which the component statements are combined.
A table presenting the truth values of the component statements together with the truth values of their compound statement, is known as the truth table. The truth table consists of a number of rows and column. Some of the initial columns contain the possible truth values of the component statements and then the truth values of the compound statements formed from the given simple statements using suitable connective.
1.4 Logical Connectives
Compound statements are made from the simple statements by using the words or phrase like "and", "or", "If ... then" and "If and only if" and they are known as logical connectives or simply connectives.
1. Conjunction
Two simple statements combined by the word "and" (or equivalent word) to form a compound statement, is known as conjunction of the given statements. The symbol used for the conjunction is ^. If p and q are two simple statements, then their conjunction is symbolized by p∧q. In case of conjunction of p and q, if p is true and q is true then p∧q is true. Otherwise p∧q is false.
Example:
p: Pabitra is an engineer,
q: Sumitra is a doctor
their conjunction is "Pabitra is an engineer and Sumitra is a doctor" and this compound statement is symbolized by p∧q.
The truth table of the conjunction of the statements p and q is presented below:
| p | q | p∧q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
2. Disjunction
Two simple statements combined by the word "or" (or an equivalent word), to form a compound statement, is known as disjunction of the given statements. The symbol used for the disjunction is v. If p and q are the two simple statements then disjunction of p and q is symbolized by p∨q. In case of disjunction of p and q, if p is true or q is true or both p and q are true then p∨q is true, otherwise p∨q is false.
Example
p: Ananda is smart
q: Arun is handsome;
their disjunction is "Ananda is smart or Arun is handsome" and this compound statement is symbolized by p∨q.
The truth table of the disjunction of p and q is presented below:
| p | q | p∨q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
3. Negation
A statement which denies the given statement is known as the negation of a given statement. The negation of a given statement is obtained by inserting the word "not" in the given statement or by adding "It is not true" or "It is not the case that" at the beginning of the given statement. The symbol used for the negation is ~. If p is the given statement, then its negation is symbolized by ~p. In case of negation, if p is true, then ~p is false and if p is false then ~p is true.
Example:
p: Krishna wears spectacle.
Then, ~p is "Krishna does not wear spectacle."
The truth table for the negation of a statement p is presented below:
| p | ~p |
|---|---|
| T | F |
| F | T |
The negation of the words "all", "some", "some ... not" and "no" are "some ... not", "no", "all" and some respectively.
Example: p: All students are laborious
~p: Some students are not laborious
4. Conditional (Implication)
Two simple statements combined by "If ... then" to form a compound statement, is known as the conditional of the given statements. The symbol used for the conditional (or implication) is ⇒. The conditional of the simple statements p and q is symbolized by p⇒q Here p is known as the antecedent and q, the consequent. In case of conditional, the conditional p⇒q is false when p is true but q is false, otherwise it is true.
Example:
p: ABC is a triangle.
q: the sum of three angles is 180°.
The conditional of p and q is "If ABC is a triangle then the sum of three angles is 180°" and this statement is symbolized by p⇒q.
The truth table of the conditional p⇒q is presented below:
| p | q | p⇒q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
5. Biconditional (Equivalence)
Two simple statements combined with "if and only if" (abbreviated as iff) to form a compound statement, is known as a biconditional. The symbol used for biconditional is ⇔. The biconditional of the statement p and q is symbolized by p⇔q which means p⇒q and q⇒p. In case of biconditional, p⇔q is true when both p and q have the same truth value and is false when p and q have different truth values.
Example:
p: Two triangles are congruent
q: corresponding sides of two triangles are equal.
Then the biconditional of p and q is "Two triangles are congruent if and only if their corresponding sides are equal." i.e. p⇔q This biconditional contains the following two cases:
- If two triangles are congruent then their corresponding sides are equal i.e. p⇒q.
- If the corresponding sides of two triangles are equal, then they are congruent. i.e. q⇒p.
p⇔q is same as p⇒q and q⇒p i.e. (p⇒q)∧(q⇒p)
The truth table of the biconditional of p and q is presented below:
| p | q | p⇔q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Following table gives the truth values of the compound statements formed with different connectives when the different truth values of the component statements are given.
| p | q | p∧q | p∨q | ~p | p⇒q |
|---|---|---|---|---|---|
| T | T | T | T | T | |
| T | F | F | T | F | |
| F | T | F | T | T | T |
| F | F | F | F | T | T |
Before discussing the laws of logic we have the following definitions:
Tautology: A compound statement which is always true, whatever may be the truth values of its components, is known as a tautology. The statement that "It is a rainy day or it is not a rainy day" is a tautology.
Contradiction: A compound statement which is always false, whatever may be the truth values of its components is known as a contradiction. The statement that "I like coffee and I don't like coffee" is a contradiction.
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