Question

\(\mathrm{p} \Rightarrow \mathrm{q} \mathrm{V} \mathrm{r}\) is false then the true values of \(\mathrm{p}, \mathrm{q}\) and \(\mathrm{r}\) are respectively. (a) \(\mathrm{F}, \mathrm{T}, \mathrm{T}\) (b) \(\mathrm{T}, \mathrm{T}, \mathrm{F}\) (c) \(\mathrm{T}, \mathrm{F}, \mathrm{F}\) (d) \(\mathrm{F}, \mathrm{F}, \mathrm{F}\)

Short answer

The short answer is: \(\mathrm{(c) \ T, F, F}\).

Step-by-step solution

Analyzing the statement pattern p → q ∨ r

Here we have a conditional with the antecedent p and the consequent as a disjunction of q and r (q ∨ r). First, let's recall the truth table of a conditional p → q: p | q | p → q T | T | T T | F | F F | T | T F | F | T Now, let's recall the truth table of a disjunction q ∨ r: q | r | q ∨ r T | T | T T | F | T F | T | T F | F | F

Making the statement pattern p → q ∨ r false

For the given statement pattern to be false, the antecedent p must be true, and the whole statement q ∨ r must be false. Looking at our disjunction truth table, we see that q ∨ r is false when both q and r are false. This leads us to the following truth values for p, q, and r: p: True q: False r: False This matches the option (c) \(\mathrm{T}, \mathrm{F}, \mathrm{F}\).

Key concepts

Truth Tables

Truth tables provide a simple yet powerful way of visualizing the results of various logical operations. They list all possible truth values of the variables involved and the corresponding outcome of the logical expression.
Consider the example conditional statement: \( p \to q \)

• If \( p \) is true and \( q \) is true, the statement is true.
• If \( p \) is true and \( q \) is false, the statement is false.
• If \( p \) is false, regardless of \( q \), the statement is true.

To provide another typical example, a disjunction such as \( q \lor r \) is only false when both \( q \) and \( r \) are false, otherwise it is true.
Truth tables thus help you systematically determine the outcome based on the truth values of individual propositions.

Conditional Statements

A conditional statement, often referred to as an "if-then" statement, takes the form \( p \to q \), which can be read as "if \( p \), then \( q \)." It's important to understand the conditions under which a conditional statement is true or false.
In logic, the statement \( p \to q \) is false only when \( p \) is true and \( q \) is false. In all other cases, it is true:

• When both \( p \) and \( q \) are true, \( p \to q \) is true.
• If \( p \) is false, the truth value of \( q \) doesn't matter; \( p \to q \) is true.

Understanding these nuances helps in constructing and analyzing conditional statements correctly.

Disjunction

Disjunction in logic is a connective used to join two statements, represented by \( \lor \), meaning "or." The expression \( q \lor r \) indicates that either \( q \) or \( r \) (or both) must be true for the disjunction to be true.
A disjunction is only false when both components are false:

• If \( q \) is true and \( r \) is false, \( q \lor r \) is true.
• If \( q \) is false and \( r \) is true, \( q \lor r \) is true.
• It is false only when both \( q \) and \( r \) are false.

Disjunction is a fundamental operation in logic that allows combining multiple conditions in a flexible manner.

Logical Reasoning

Logical reasoning is the process of using a structured, step-by-step approach to reach a conclusion based on a set of premises or known facts. By applying the principles of logic, you can decipher complex statements and solve problems systematically.
Consider the compound statement \( p \to q \lor r \):

• The aim is to determine the truth value of \( p \to q \lor r \).
• Recognize that for the statement \( p \to q \lor r \) to be false, \( p \) must be true, and \( q \lor r \) must be false.

The use of truth tables and understanding of conditional and disjunction statements is pivotal in executing logical reasoning effectively. This makes it easy to identify the correct set of truth values for complex logical statements.