Logic
Symbolic logic
uses mathematical symbols to represent sentence and is used to determine the
truth or falsity of the sentence.
It is the systematic study of process of reasoning.
Statement: A statement or a proposition is a declarative sentence that is either true
or false but not both true and false.
Statements are generally denoted by p, q, r,
s, ……
Examples of Statement:
Trishuli is in Nuwakot.
3 + 4 = 7
8 + 4 = 9
A triangle has three sides.
Some sentences which are not statements are
as follows:
How wonderful!
Close the door.
Who is the headteacher of Trishuli School?
Go to your school.
Open sentences
Example:
x + 3 = 6
y × 4 = 20
………… is the capital city of Nepal.
These types of sentences are open sentences and these
sentences are not statements.
Simple Statement
The statement expressing single complete
thought is called a simple or prime statement. In other word, a statement whose
truth value does not depend on another statement is said to be simple.
4 is less than 10.
3 is a prime number.
Mt. Everest is in Nepal.
Compound Statement
Statements that are the combinations of two
or more simple statements are called compound statements.
A compound statement is formed by combining
two or more statements with a connective such as and, or, not, if …., then, if
and only if (iff) etc.
Examples.
3 is an odd number and 4 is an even number.
3 – 1 = 2 and 4>2
Truth values
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 to
it is true or false.
Truth Table
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.
Logical Connectives
The word used to combine two or more
statements is called logical connectives. The connectives used to form a
compound statement are presented in table below. These connectives are
frequently called fundamental operators.
Conjunction
Two simple statements combined by the word "and" to form a
compound statement is known as conjunction of the given statements.
Let p and q are two simple statements. A statement of the form " p and
q" is called conjunction. It is denoted by p ∧ q.
Eg. Triangle has three sides and square has four
sides.
5 +3 = 8 and 6 >1
Truth table of Conjunction
|
p |
q |
p ∧ q |
|
T |
T |
T |
|
T |
F |
F |
|
F |
T |
F |
|
F |
F |
F |
A conjunction p ∧ q is true if both p and q are true. it is false in other cases.
Disjunction
Two simple statements combined by the word "or" to form a
compound statement is known as disjunction of the given statements.
Let p and q are two simple statements. A statement of the form " p or
q" is called conjunction. It is denoted by p ∨ q.
eg. 2 is an even or 3 is an odd number.
Truth table of Disjunction
p | q | p ∨q |
T | T | T |
T | F | T |
F | T | T |
F | F | F |
Let p be a statement. A statement of the form " not p" is called Negation of p. It is denoted by "∼P".
Truth Table
|
p |
∼ p |
|
T |
F |
|
F |
T |
Some words and its negation
|
Words |
Its Negation |
|
All |
Some …….not |
|
Some |
No |
|
Some not |
All |
|
No |
Some |
Example :
Statement : All Students are honest.
Negation : Some students are not honest.
Statement : Some students are honest.
Negation : No students are honest.
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