Logical operators are used to construct mathematical statements having one or more propositions by combining the propositions. The combined proposition is called compound Proposition. The truth table is used to get the relationship between truth values of propositions. Here we present the logical operators along with their behavior in truth table:
1. Negation (NOT):
Examples: Negation of proposition “Today is Sunday” is “It is not the case that today is Sunday” or “Today isn’t Sunday” or “It is not Sunday today”
p: The summer in Terai is very hot.
¬p: The summer in Terai isn’t very hot.
2. Conjunction (AND):
Given two propositions p and q, the proposition “p and q” denoted by p∧q is the proposition that is true whenever both the propositions p and q are true, false otherwise. The proposition that is obtained by the use of “and” operator is also called conjunction of p and q.
Example: If we have propositions p = “Ram is intelligent” and q = “Ram is diligent”, the conjunction of p and q is Ram is intelligent and diligent. This proposition is true only when Ram is intelligent and he is diligent also, false otherwise.
Let, p: Today is Saturday.
q: Today is holiday
Then, p∧q: Today is Saturday and holiday.
3. Disjunction (OR):
Given two propositions p and q, the proposition “p or q” denoted by p∨q is the proposition that is false whenever both the propositions p and q are false, true otherwise. The proposition that is obtained by the use of “or” operator is also called disjunction of p and q.
Example: If we have propositions p = “Ram is intelligent” and q = “Ram is diligent”, the disjunction of p and q is Ram is intelligent or diligent. This proposition is false only when Ram is not intelligent and not diligent, true otherwise.
Let, p: It is cold.
q: It is raining.
Then, p∨q: It is cold or raining.
4. Exclusive or (XOR):
Given two propositions p and q, the proposition exclusive or of p and q denoted by p⊕q is the proposition that is true whenever only one of the propositions p and q is true, false otherwise. As opposed to the disjunction above which is inclusive the general meaning of the English sentence can be used to know whether the “or” used is inclusive or exclusive.
Example: If we have propositions p = “Ram drinks coffee in the morning” and q = “Ram drinks tea in the morning”, the exclusive or of p and q is Ram drinks coffee or tea in the morning. When you buy a car from XYZ company, you get $250 cashback or accessories worth $250.
5. Implication (Conditional):
Given two propositions p and q, the proposition implication p→q is the proposition that is false when p is true and q is false, true otherwise. Here p is called “hypothesis” or “antecedent” or “premise” and q is called “conclusion” or “consequence”. If p is false then it doesn’t matter what will be the truth value of q, p→q is always true.
We use different terminologies to express p→q like:
“if p, then q”, q is consequence of p”, “p is sufficient for q”, “q if p”, “q is necessary for p”, “q follows from p”, “if p, q”, “p implies q”, “p only if q”, “q whenever p”, “q provides p”, “q unless ¬p”
Example: p = “Today is Sunday” q = “It is hot” then, the implication can be “If today is Sunday then it is hot today” or “Today is Sunday only if it is hot today”.
If you try hard for your exam, then you will succeed.
p: You try hard for your exam
q: you will succeed
Therefore, p→q
Examples logical connectives:
Let, p = “It rained last night”
q = “The sprinkles came on last night”
r = “The lawn was wet this morning”
Translate the following into English:
¬p, r∧¬p and ¬r∨p∨q
Let p, q and r be the propositions with truth values T, F, T respectively.
Evaluate the following:
¬r∨¬(p∧q) and ¬(p∨q)∧(¬r∨q)
Determine whether each of these conditional statements are True or False.
If 1+1=3, then dogs can fly.
If 1+1=2, then dogs can fly.
If monkey can fly, then 1+1=3.
If 1+1=2, then 2+2=5.
If Kathmandu is capital of Nepal then Delhi is capital of India.
Inverse, Converse & Contrapositive:
Some of the related implications formed from p→q are:
Inverse of Implication:
When we add “not” to the hypothesis and conclusion of implication p→q then it becomes ¬p→¬q, which is known as inverse of p→q.
p→q: If it is raining, then the road is muddy.
¬p→¬q: If it isn’t raining, then the road isn’t muddy.
Converse of Implication:
When we flip/interchange the hypothesis and conclusion of implication p→q then the result becomes q→p, which is known as converse of given implication.
p→q: If it is raining, then the road is muddy.
q→p : If the road is muddy then it is raining .
Contrapositive of Implication:
When we flip/interchange the hypothesis and conclusion of inverse statement of implication p→q then the resulting statement ¬q→¬p is known as contrapositive.
p→q: If it is raining, then the road is muddy.
¬q→¬p : If the road isn’t muddy then it isn’t raining .
Example:
Write the inverse, converse and contrapositive of the following statements:
•If two angles are congruent, then they have same measures.
•If quadrilateral is rectangle then it has two pairs of parallel sides.
•I go to swimming, whenever it is sunny.
•When I stay up late, it is necessary that I sleep until noon.
6. Biconditional:
Given propositions p and q, the biconditional p↔q is a proposition that is true when p and q have same truth values. Alternatively p↔q is true whenever both p→q and q→p are true.
Some of the terminologies used for biconditional are: “p if and only if q”, “if p then q, and conversely”, “p is necessary and sufficient for q”
Example:
p = “Today is Sunday” q = “It is hot” then biconditional statement is, “Today is Sunday if and only if it is hot today”.
Examples of Biconditional:
Determine whether the following propositions are True or False.
•2+3=5 if and only if 2+1=3.
•1+1=2 if and only if pigs can fly.
•If 1+1=3, then God exist and conversely.
Precedence of Logical Operators:
- Negation
- Conjunction
- Disjunction
- Implication
- Biconditional