Question

\(\mathrm{p} \Rightarrow(\mathrm{q} \Rightarrow \mathrm{p})\) is equivalent to (a) \(p \Rightarrow(p \Leftrightarrow q)\) (b) \(p \Rightarrow(p \Rightarrow q)\) (c) \(p \Rightarrow(p \vee q)\) (d) \(p \Rightarrow(p \wedge q)\)

Short answer

The statement \(p \Rightarrow (q \Rightarrow p)\) is equivalent to option (c) \(p \Rightarrow (p \vee q)\).

Step-by-step solution

Write the truth table for the given statement

To write the truth table for the given statement \(\mathrm{p} \Rightarrow(\mathrm{q} \Rightarrow \mathrm{p})\), we will first write down all possible combinations of truth values for \(\mathrm{p}\) and \(\mathrm{q}\), and then determine the truth value of the entire statement for each combination.

Write truth tables for the given options

Next, we will write the truth tables for each of the given options (a) to (d).

Compare the truth tables

Now we will compare the truth table of the given statement with the truth tables of the options. Here are the truth tables for the given statement and the options: Given statement \(\mathrm{p} \Rightarrow(\mathrm{q} \Rightarrow \mathrm{p})\) | p | q | q → p | p → (q → p) | |---|---|-------|-------------| | T | T | T | T | | T | F | T | T | | F | T | F | T | | F | F | T | T | Option (a) \(\mathrm{p} \Rightarrow(\mathrm{p} \Leftrightarrow \mathrm{q})\) | p | q | p ↔ q | p → (p ↔ q) | |---|---|-------|-------------| | T | T | T | T | | T | F | F | F | | F | T | F | T | | F | F | T | T | Option (b) \(\mathrm{p} \Rightarrow(\mathrm{p} \Rightarrow \mathrm{q})\) | p | q | p → q | p → (p → q) | |---|---|-------|-------------| | T | T | T | T | | T | F | F | F | | F | T | T | T | | F | F | T | T | Option (c) \(\mathrm{p} \Rightarrow(\mathrm{p} \vee \mathrm{q})\) | p | q | p ∨ q | p → (p ∨ q) | |---|---|-------|-------------| | T | T | T | T | | T | F | T | T | | F | T | T | T | | F | F | F | T | Option (d) \(\mathrm{p} \Rightarrow(\mathrm{p} \wedge \mathrm{q})\) | p | q | p ∧ q | p → (p ∧ q) | |---|---|-------|-------------| | T | T | T | T | | T | F | F | F | | F | T | F | T | | F | F | F | T | By comparing the truth table of the given statement with the truth tables of the options, we can see that the truth table of option (c) \(\mathrm{p} \Rightarrow(\mathrm{p} \vee \mathrm{q})\) is identical to the truth table of the given statement \(\mathrm{p} \Rightarrow(\mathrm{q} \Rightarrow \mathrm{p})\). Therefore, the statement \(\mathrm{p} \Rightarrow(\mathrm{q} \Rightarrow \mathrm{p})\) is equivalent to option (c) \(\mathrm{p} \Rightarrow(\mathrm{p} \vee \mathrm{q})\).

Key concepts

Truth Tables

To fully comprehend the foundation of logical equivalence, we begin by looking at truth tables. Truth tables are a systematic way to list all possible combinations of truth values for a given set of propositions and their logical connectives. They help in determining the truth or falsity of complex statements based on their components.

In our example, we visualized the truth values of \(\mathrm{p} \rightarrow(\mathrm{q} \rightarrow \mathrm{p})\) and the provided options by considering every possible truth assignment to \(\mathrm{p}\) and \(\mathrm{q}\). Truth tables serve as a crucial tool for comparing different logical expressions. They are especially useful for verifying logical equivalence as we can clearly observe which statements produce the same output for every input, thereby confirming equivalence.

Logical equivalence can readily be identified through truth tables as we search for columns with identical truth values. Thus, systematically creating and comparing truth tables is an indispensable skill to have when working with logical expressions.

Conditional Statements

Diving deeper, conditional statements are expressions in logic that are structured in an 'if-then' form, symbolized as \(\mathrm{p} \rightarrow \mathrm{q}\). In the realm of logical operations, the conditional, also known as implication, communicates that if the first proposition \(\mathrm{p}\) is true, then the second proposition \(\mathrm{q}\) should also be true. Interestingly, if \(\mathrm{p}\) is false, the whole conditional is true regardless of the truth value of \(\mathrm{q}\).

In our problem, we explored statements involving nested conditionals, such as \(\mathrm{p} \rightarrow(\mathrm{q} \rightarrow \mathrm{p})\) where the truth of \(\mathrm{p}\) guarantees the truth of the expression \(\mathrm{q} \rightarrow \mathrm{p}\). Understanding the intricacies of such statements is crucial because it helps decode complex logical constructs into simpler, more intuitive components. By breaking down the conditional statement into parts, we can employ truth tables to evaluate each segment and determine the validity of the whole expression.

Logical Connectors

Finally, logical connectors, also known as logical operators or connectives, are the building blocks in constructing complex logical expressions. Common connectors include 'and' (conjunction: \(\wedge\)), 'or' (disjunction: \(\vee\)), 'not' (negation: \(eg\)), and 'if...then' (conditional: \(\rightarrow\)). They define how the truth values of individual propositions interact with one another within a statement.

These connectors follow specific rules that dictate the resultant truth values. For example, in conjunction \(\mathrm{p} \wedge \mathrm{q}\), both \(\mathrm{p}\) and \(\mathrm{q}\) must be true for the entire statement to be true. Meanwhile, in disjunction \(\mathrm{p} \vee \mathrm{q}\), if either \(\mathrm{p}\) or \(\mathrm{q}\) is true, the whole statement is considered true. These rules were implicitly followed when constructing the truth tables in our exercise.

Recognizing the function and effects of each logical connector is imperative when learning how to construct and interpret logical expressions. By mastering their use, one can solve complex logical problems by breaking them down into a series of simpler, more manageable parts.