Logical connectives are the operators used to join two or more atomic propositions (operands). The joining should be done in a way that the logic and truth value of the obtained compound proposition is dependent on the input atomic propositions and the connective used.
- Conjunction
A proposition “A ∧ B” with connective ∧ is known as conjunction of A and B. It
is a proposition (or operation) which is true only when both the constituent
propositions are true. Even if one of the input propositions is false then the
output is also false. It is also referred to as AND-ing the propositions. Example:
Ram is a playful boy and he loves to play football. It can be written as:
A = Ram is a playful boy.
B = Ram loves to play football.
A ∧ B = Ram is a playful boy and he loves to play football.
2. Disjunction
A proposition “A ∨ B”with connective ∨ is known as disjunction of A and B. It
is a proposition (or operation) which is true when at least one of the constituent
propositions are true. The output is false only when both the input propositions
are false. It is also referred to as OR-ing the propositions. Example:
I will go to her house or she will come to my house. It can be written as:
A = I will go to her house.
B = She will come to my house.
A ∨ B = I will go to her house or she will come to my house.
3. Negation
The proposition ¬ A (or ~A) with ¬ (or ~) connective is known as negation of
A. The purpose of negation is to negate the logic of given proposition. If A is
true, its negation will be false, and if A is false, its negation will be true.
Example:
University is closed. It can be written as:
A = University is closed.
¬ A = University is not closed.
4. Implication
The proposition A → B with → connective is known as A implies B. It is also
called if-then proposition. Here, the second proposition is a logical consequence
of the first proposition. For example, “If Mary scores good in examinations, I
will buy a mobile phone for her”. In this case, it means that if Mary scores
good, she will definitely get the mobile phone but it doesn’t mean that if she
performs bad, she won’t get the mobile phone. In set notation, we can also say
that A ⊆ B i.e., if something exists in the set A, then it necessarily exists in the
set B. Another example:
If you score above 90%, you will get a mobile phone.
A = You score above 90%.
B = You will get a mobile phone.
A → B = If you score above 90%, you will get a mobile phone.
5. Bi-conditional
A proposition A ⟷ B with connective ⟷ is known as a biconditional or if-andonly-if proposition. It is true when both the atomic propositions are true or both
are false. A classic example of biconditional is “A triangle is equivalent if and
only if all its angles are60° each”. This statement means that if a triangle is an
equivalent triangle, then all of its angles are 60° each. There is one more
associated meaning with this statement which means that if all the interior
angles of a triangle are of 60° each then it's an equivalent triangle. Example:
You will succeed in life if and only if you work hard.
A = You will succeed in life.
B = You work hard.
A ⟷ B = You will succeed in life if and only if you work hard.
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