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VOOZH | about |
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VOOZH | about |
Propositional logic is a branch of mathematics that studies the logical relationships between propositions (or statements, sentences, assertions) taken as a whole and connected via logical connectives.
It focuses on how these propositions relate to each other through logical connectives such as AND, OR, NOT, IF…THEN, etc.
Examples of Propositions
| Statement | Truth Value |
|---|---|
| The sun rises in the East and sets in the West. | True |
| 1 + 1 = 2 | True |
| ‘b’ is a vowel. | False |
All of the above are propositions because each has a definite truth value.
Some sentences are not propositions because they don’t have a definite truth value or may vary depending on context:
In propositional logic, logical connectives are symbols used to build compound propositions from atomic ones.
In propositional logic, propositions are statements that can be evaluated as true or false. They are the building blocks of more complex logical statements. Two main types of propositions:
Since we need to know the truth value of a proposition in all possible scenarios, we consider all the possible combinations of the propositions that are joined together by Logical Connectives to form the given compound proposition. This compilation of all possible scenarios in a tabular format is called a truth table. Most Common Logical Connectives-
If p is a proposition, then the negation of p is denoted by ¬p, which, when translated to simple English, means "It is not the case that p" or simply "not p." The truth value of -p is the opposite of the truth value of p. The truth table of -p is:
| p | ¬p |
|---|---|
| T | F |
| F | T |
Example, Negation of "It is raining today" is "It is not the case that it is raining today" or simply "It is not raining today."
For any two propositions p and q, their conjunction is denoted by p∧q, which means "p and q." The conjunction p∧q is True when both p and q are True, otherwise False. The truth table of p∧q is:
| p | q | p ∧ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Example, Conjunction of the propositions p—"Today is Friday" and q—"It is raining today," p∧q, is "Today is Friday and it is raining today." This proposition is true only on rainy Fridays and is false on any other rainy day or on Fridays when it does not rain.
For any two propositions p and q, their disjunction is denoted by p∨q, which means "p or q." The disjunction p∨q is True when either p or q is True, otherwise False. The truth table of p∨q is:
| p | q | p ∨ q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Example, Disjunction of the propositions p—"Today is Friday" and q—"It is raining today," p∨q, is "Today is Friday or it is raining today." This proposition is true on any day that is a Friday or a rainy day (including rainy Fridays) and is false on any day other than Friday when it also does not rain.
For any two propositions p and q, their exclusive or is denoted by p⊕q, which means "either p or q but not both." The exclusive or p⊕q is True when either p or q is True, and False when both are true or both are false. The truth table of p⊕q is:
| p | q | p ⊕ q |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
Example, Exclusive or of the propositions p—"Today is Friday" and q—"It is raining today," p⊕q is "Either today is Friday or it is raining today, but not both." This proposition is true on any day that is a Friday or a rainy day (not including rainy Fridays) and is false on any day other than Friday when it does not rain or rainy Fridays.
For any two propositions, p and q, the statement "if p then q" is called an implication, and it is denoted by p→q. In the implication p→q, p is called the hypothesis or antecedent or premise, and q is called the conclusion or consequence. The implication is that p→q is also called a conditional statement. The implication is false when p is true and q is false; otherwise, it is true. The truth table of p→q is:
| p | q | p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
One might wonder why p→q is true when p is false. This is because the implication guarantees that when p and q are true, then the implication is true. But the implication does not guarantee anything when the premise p is false. There is no way of knowing whether or not the implication is false since p did not happen. This situation is similar to the "Innocent until proven Guilty" stance, which means that the implication p→q is considered true until proven false. Since we cannot call the implication p→q false when p is false, our only alternative is to call it true.
This follows from the Explosion Principle which says: "A False statement implies anything." Conditional statements play a very important role in mathematical reasoning; thus, a variety of terminology is used to express p → q, some of which are listed below.
"If p, then "q"p is sufficient for q""q when p""a necessary condition for p is q""p only if q""q unless ≠p""q follows from p"
Example, "If it is Friday, then it is raining today" is a proposition that is of the form p→q. The above proposition is true if it is not Friday (premise is false) or if it is Friday and it is raining, and it is false when it is Friday but it is not raining.
For any two propositions p and q, the statement "p if and only if (iff) q" is called a biconditional, and it is denoted by p↔q. The statement p↔q is also called a bi-implication. p↔q has the same truth value as (p→q) ∧ (q→p). The implication is true when p and q have the same truth values and is false otherwise. The truth table of p→q is:
| p | q | p ↔ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Some other common ways of expressing p↔q are:
"p is necessary and sufficient for q" "if p then q, and conversely" "p if q"
Example, "It is raining today if and only if it is Friday today" is a proposition that is of the form p↔q. The above proposition is true if it is not Friday and it is not raining or if it is Friday and it is raining, and it is false when it is not Friday or it is not raining.
1) Consider the following statements:
Which one of the following about L, M, and N is CORRECT?
(A) Only L is TRUE.
(B) Only M is TRUE.
(C) Only N is TRUE.
(D) L, M, and N are TRUE.
Solution:
Let a and b be two proposition
a: Good Mobile phones.
b: Cheap Mobile Phones.
P and Q can be written in logic as
P: a-->~b
Q: b-->~a.
Truth Table
a b ~a ~b P Q
T T F F F F
T F F T T T
F T T F T T
F F T T T T
it clearly shows P and Q are equivalent.
so option D is Correct
2)Which one of the following is not equivalent to p <-> q
(A)
(B)
(C)
(D)
Conjunction of p and q, denoted by p∧q, is the proposition ‘p and q.'. The conjunction p ∧ q is True, when both p and q is True. Disjunction of p and q, denoted by p∨q, is the proposition ‘p or q.'. The disjunction p∨q is False when both p and q is False.
Logical Implication - It is a type of relationship between two statements or sentences. Denoted by ‘p → q.'. The conditional statement p → q is false when p is true and q is false and true otherwise. i.e., p → q = ¬p ∨ q
Bi-Condition A bi-conditional statement is a compound statement formed by combining two conditionals under “and.” Bi-conditionals are true when both statements have the exact same truth value.
Solution:
A biconditional is true when both propositions have the same truth value: p↔q≡(p∧q)∨(¬p∧¬q)
Option (D) matches this directly.
Option (A): (¬p∨q) ∧ (p∨¬q) is equivalent to (¬p∨q) ∧ (¬q∨p), which is p ↔ q.
Option (B): q→p is ¬q∨p, so (B) is the same as (A).
Option (C): (¬p∧q) ∨ (p∧¬q)
is true exactly when p and q have opposite truth values: p⊕q = ¬(p↔q)
Only option which is not equivalent to p↔q is option (C). So, option (C) is correct.
∧ q) which is Option (D)
Question 1: Given:
Check if the statement is logically equivalent to Justify with truth table
Question 2: Simplify (p∨q) ∧(¬p∨q) to an equivalent expression using logical laws.
Question 3: Let:
Which of the following is true?
(A) P and Q are equivalent, but not R
(B) P and R are equivalent, but not Q
(C) All three are equivalent
(D) None are equivalent
Question 4: Given P: p→(q∨r) and Q: (p→q)∨(p→r). Are P and Q logically equivalent? Justify using a truth table.
