In mathematics, the study of logic is fundamental to understanding and constructing valid arguments. A significant aspect of logical reasoning involves the use of connectives, which are words or symbols that combine or alter simple statements to create more complex statements known as compound statements.
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The words which combine or change simple statements to form new statements or compound statements are called Connectives. The basic connectives (logical) conjunction corresponds to the English word ‘and’, disjunction corresponds to the word ‘or’, and negation corresponds to the word ‘not’.
Name of Connective | Connective Word | Symbol |
Conjunction | And | ⋀ |
Disjunction | Or | ⋁ |
Negation | Not | 〜 |
Conditional | ‘if-then' or 'implication' | ➝ or ⇒ |
Biconditional | ‘If and only if' or 'double implication' | ↔️ or ⇔ |
Negation of a Statement
or "It is false that".
The negation of a statement
The negation of this statement is
Or,
Or,
Conjunction
If two simple statements
For example,
The statement can be broken into two component statements as
The compound statement with 'And' is true if all its component statements are true.
The component statement with 'And' is false if any of its component statements are false.
Note:
A statement with "And" is not always a compound statement.
For example,
Here the word "And" refers to two things - alcohol and water.
Disjunction
If two simple statements
The statement can be broken into two component statements as
The compound statement with 'or' is true if any of its component statements are true.
The component statement with 'or' is false if all of its component statements are false.
Types of OR statements
1. Inclusive OR: If
Eg, 'Bangalore is in Karnataka or India'
Here both component statements 'Bangalore is in Karnataka' and ' 'Bangalore is in India' can be true simultaneously. Hence this compound statement has an inclusive OR.
2. Exclusive OR: If
Eg, 'Bangalore is in Karnataka or Maharashtra'
Here both component statements 'Bangalore is in Karnataka' and ' 'Bangalore is in Maharashtra' cannot be true simultaneously. Hence this compound statement has an exclusive OR.
Conditional Statement
Then the sentence "if
The conditional statement
1.
2.
3.
4.
5.
The statement
The Biconditional Statement
If two statements
ne segments are congruent if and only if they are of equal length'
It is a combination of two conditional statements, "if two line segments are congruent then they are of equal length" and "if two line segments are of equal length then they are congruent". Which means
A biconditional is true if and only if both the statements
Also a biconditional is true if
Example 1: What does symbol "
1)OR
2)AND
3) NOT
4) IF
Solution
Conjunction -
Symbol "
Symbol "^" depicts AND
Example 2: Which one is NOT an example of an AND conjunction?
1)
2)
3)
4)
Solution
'And' Conjunction -
Normally the conjunction 'and' is used between two statements which have some kind of relation but in logic, it can be used even if there is no relation between the statements.
"Sam opened the closet and took out clothes" is not an example of an AND conjucton as "and " is used in a different sense here.
Example 3: Find the entire truth set of
1)
2)
3)
4) none of these
Solution
Logic Connectivity -
Note:
Example 4: How will you prove that
1) put
2) put
3) put
4) none of these
Solution
Logic Connectivity -
Truth Value of a Statement
As we know that a statement is either true or false. The truth or falsity of a statement is called truth value.
If the statement is true, then truth value is " T "
If the statement is false, then truth value is "
It can be done by contradiction i.e assuming
Example 5: Let
Which of the following statements is the negation of the statement
1) There is a rational number
2) There is no rational number
3) Every rational number
4)
Solution
Logic Connectivity -
Negation of a Statement
The negation of a statement
The negation of this statement is
Or,
Or,
The truth value of negation of a statement is always opposite to the truth value of the original statement.
We write it up as
Summary
Logical connectives are essential tools in mathematics and logic, allowing for the construction of complex statements from simpler ones. By understanding how conjunction, disjunction, negation, conditional, and biconditional connectives function, one can form precise logical arguments and engage in rigorous reasoning. These connectives not only help in mathematical proofs but also in everyday logical thinking, making them fundamental components of clear and effective communication.
The words which combine or change simple statements to form new statements or compound statements are called Connectives.
The connectives are conjuction(AND), disjunction(OR), negation(NOT), conditional and biconditional connectives.
If two simple statements
If two simple statements
If two statements
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