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Introduction:
The conditional and biconditional statements are the statements put in p and q format. Any statement put in the format “If p, then q” is called a conditional statement. It is written as p → q. The conditional statement is also known as implication.It can also be written as “p implies q.” The arrow follows the implication logic expressed in a conditional statement. The p component is premise or antecedent, and the q component is known as conclusion or consequent.
On the other hand, biconditional statements are the statements that are written in the form of “p if and only if q.” Here, the p and q are known as the basic statements. The biconditional statements are the conjunctions of the conditional statements with the converse. It is written as p ↔ q.
Conditional Statements
The conditional statements are in the form of p → q with the premise p. The statement number is divided by two and the conclusion by q. This way, the number is even. The implication of the conditional statement p → q is only false when q is false and p is true. In all the other implications, it is always true. In the implication, p is known as antecedent or hypothesis, and q is known as consequent or conclusion.
The truth table of the conditional statements is as follows:
p | q | p → q |
T | T | T |
T | F | F |
F | T | T |
F | F | T |
The example of the conditional statements are as follows:
- If x=y and y=z, then x=z.
- If I get money, then I will buy a watch.
Variations in Conditional Statements
There are three variations in the conditional statements, and they are as follows:
- Inverse – The implication of ~p→ ~q is known as the inverse of p → q.
- Contrapositive – The implication of ~q→ ~p is known as contrapositive of p →q
- Converse – The implication of q→ p is known as the converse of p → q.
Example of Conditional Statements
- Show that p → q and its contrapositive ~q → ~p are equivalent logically.
Sol: The truth table of the propositions mentioned in the questions are as follows:
p | q | ~p | ~q | P → q | ~q→ ~p |
T | T | F | F | T | T |
T | F | F | T | F | F |
F | T | T | F | T | T |
F | F | T | T | T | T |
As you can see that the values in both the cases are the same, which means that both the propositions are equivalent to each other.
Biconditional Statements
The biconditional statements are written as p ↔ q. It is also known as equivalence and is often written as “p is equivalent to q.”
(𝑝 → 𝑞) ∧ (𝑞 → 𝑝)
≡ (𝑝 → 𝑞) ∧ (𝑝 ← 𝑞)
≡ 𝑝 ↔ 𝑞
Symbolically it is, p ≡ q.
The truth table of the biconditional statements is as follows:
p | q | p ↔ q |
T | T | T |
T | F | F |
F | T | F |
F | F | T |
So, it shows that the biconditional statements are true when p and q are both either false or true. The condition is “p if and only if q”.
Bi conditional statement examples are as follows:
- A polygon is called hexagon if and only if it has 6 sides.
- You will pass the exam if and only if you will work very hard for it.
Bi conditional Statement Example:
- Prove that p ↔ q is equivalent to (p →q) ∧ (q → p)
Sol: The truth table of the biconditional statement is as follows:
p | q | p ↔ q |
T | T | T |
T | F | F |
F | T | F |
F | F | T |
p | q | p→ q | q→ p | (p→ q) ^ (q→ p) |
T | T | T | T | T |
T | F | F | T | F |
F | T | T | F | F |
F | F | T | T | T |
Principle of Duality
The two formulas A1 and A2 are considered as the duals of each other, only if the other can be replaced with ∧ (AND) by ∨ (OR) or ∨ (OR) by ∧ (AND). Once it’s done, the formula might contain F (False) or T (True). So, if it includes T, replace it with F and if it includes F, replace it with T to make a dual.
- The ∨ and ∧ are dual of each other.
- NAND and NOR are also dual of each other
- If a formula has a valid proposition, it’s a dual of each other.
Equivalence of Proposition:
The propositions are considered equivalent if they fall under the category of the same circumstances.
The following table shows the algebra of propositions:
Idempotent Laws | (i) p ∨ p ≅ p | (ii) p ∧ p ≅ p |
Associative laws | (i) (p ∨ q) ∨ r ≅ p∨ (q ∨ r) | (ii) (p ∧ q) ∧ r ≅ p ∧ (q ∧ r) |
Commutative laws | (i) p ∨ q ≅ q ∨ p | (ii) p ∧ q ≅ q ∧ p |
Distributive laws | (i) p ∨ (q ∧ r) ≅ (p ∨ q) ∧ (p ∨ r) | (ii) p ∧ (q ∨ r) ≅ (p ∧ q) ∨ (p ∧ r) |
Identity laws | (i)p ∨ F ≅ p (iv) p ∧ F≅F | (ii) p ∧ T≅ p (iii) p ∨ T ≅ T |
Involution laws | (i) ~~p ≅ p | |
Complement laws | (i) p ∨ ~p ≅ T | (ii) p ∧ ~p ≅ T |
DeMorgan’s laws | (i) ~(p ∨ q) ≅ ~p ∧ ~q | (ii) ~(p ∧ q) ≅~p ∨ ~q |
Conclusion
To conclude, we can say that the conditional and biconditional statements are the statements that are put in p and q format. Any statement that is put in the format “If p, then q”, is called a conditional statement. It is written as p→ q. On the other hand, the biconditional statements are written as p ↔ q. It is also known as equivalence and is often written as “p is equivalent to q”. Both the statements are essentially important for the students to study for their class.
