Proving Tautologies A Deep Dive Into Propositional Logic

In the realm of mathematical logic, a tautology holds a position of paramount importance. A tautology, in simple terms, is a compound statement that is always true, irrespective of the truth values of its constituent propositions. To demonstrate that a given proposition is indeed a tautology, we often employ truth tables or logical equivalences. In this article, we will delve into proving that the following propositions are tautologies, providing a comprehensive exploration of each case:

(i) (p → q) ↔ (¬q → ¬p) (ii) p ∧ (p ∨ q) ↔ p (iii) p → (q ∨ r) ↔ (p → q) ∨ (p → r)

Each proposition will be meticulously dissected, revealing the underlying logical structure and demonstrating why it invariably evaluates to true. Understanding tautologies is crucial in various fields, including computer science, philosophy, and mathematics, as they form the bedrock of logical reasoning and proof techniques.

(i) Proving (p → q) ↔ (¬q → ¬p) is a Tautology

To prove that (p → q) ↔ (¬q → ¬p) is a tautology, we will construct a truth table. This method systematically evaluates the truth value of the proposition for all possible combinations of truth values of the individual propositions p and q. The proposition (p → q) represents a conditional statement, read as "if p, then q". The symbol "¬" denotes negation, meaning "not". Thus, ¬q is the negation of q, and ¬p is the negation of p. The symbol "↔" represents a biconditional statement, which is true if and only if both sides have the same truth value.

Let's break down the components of the proposition and build the truth table step by step. First, we need to consider all possible truth values for p and q, which are:

• p = True, q = True
• p = True, q = False
• p = False, q = True
• p = False, q = False

Next, we determine the truth values of ¬q and ¬p based on the values of q and p, respectively. The negation of a true statement is false, and the negation of a false statement is true.

Now, we evaluate the truth values of the conditional statements (p → q) and (¬q → ¬p). A conditional statement is only false when the antecedent (the statement before the "→") is true and the consequent (the statement after the "→") is false. In all other cases, the conditional statement is true. So,

• (p → q) is true unless p is true and q is false.
• (¬q → ¬p) is true unless ¬q is true and ¬p is false.

Finally, we evaluate the truth value of the biconditional statement (p → q) ↔ (¬q → ¬p). This statement is true if and only if (p → q) and (¬q → ¬p) have the same truth value. In other words, it's true when both are true or both are false.

As we can see from the truth table, the column for (p → q) ↔ (¬q → ¬p) contains only True values. This definitively proves that the proposition (p → q) ↔ (¬q → ¬p) is a tautology. This particular tautology is also known as the law of contraposition, a fundamental principle in logic and mathematics. The law of contraposition states that a conditional statement is logically equivalent to its contrapositive. This equivalence is widely used in mathematical proofs and logical arguments. For example, when proving a theorem, it might be easier to prove its contrapositive, which automatically proves the original theorem as well. In essence, the contrapositive provides an alternative way to express the same logical relationship, offering flexibility and strategic advantages in reasoning and problem-solving. Understanding and applying the law of contraposition is crucial for anyone working with formal logic and mathematical proofs, allowing for more efficient and effective logical deduction.

(ii) Proving p ∧ (p ∨ q) ↔ p is a Tautology

To demonstrate that p ∧ (p ∨ q) ↔ p is a tautology, we will again employ a truth table. This method systematically examines the truth value of the proposition for all possible truth assignments to the variables p and q. The symbol "∧" represents the logical conjunction, often read as "and". The symbol "∨" represents the logical disjunction, often read as "or". The biconditional symbol "↔" remains the same, signifying that the statement is true if and only if both sides have the same truth value.

Let's start by outlining the components of the proposition and constructing the truth table step by step. The variables p and q can each take on the values True or False, giving us the following possible combinations:

• p = True, q = True
• p = True, q = False
• p = False, q = True
• p = False, q = False

Next, we need to determine the truth values of the expressions within the proposition. Let's begin with (p ∨ q). The disjunction (p ∨ q) is true if at least one of p or q is true. It is only false if both p and q are false.

Now, let's evaluate p ∧ (p ∨ q). The conjunction p ∧ (p ∨ q) is true only if both p and (p ∨ q) are true. If either p or (p ∨ q) is false, then the conjunction is false.

Finally, we evaluate the truth value of the biconditional statement p ∧ (p ∨ q) ↔ p. As we previously established, a biconditional statement is true if and only if both sides have the same truth value. In this case, the biconditional is true when p ∧ (p ∨ q) and p have the same truth value, meaning both are true or both are false.

Here is the completed truth table:

The final column in the truth table, representing p ∧ (p ∨ q) ↔ p, contains only True values. This conclusively demonstrates that the proposition p ∧ (p ∨ q) ↔ p is a tautology. This specific tautology is an example of the absorption law in propositional logic. The absorption law illustrates how one part of a compound proposition can