Problem 50

Question

Let $t$ be a tautology and $p$ an arbitrary proposition. Find the truth value of each. $$(p \wedge t) \rightarrow p$$

Step-by-Step Solution

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Answer

The truth value of $(p \wedge t) \rightarrow p$ is True (T) for all possible truth assignments of the arbitrary proposition $p$, as the tautology $t$ is always True (T).

1Step 1: Define the given expressions

We are given the tautology $t$ and the arbitrary proposition $p$. We are tasked with finding the truth value of $(p \wedge t) \rightarrow p$.

2Step 2: Construct the truth table

To find the truth value of the compound proposition $(p \wedge t) \rightarrow p$, we must consider all possible truth assignments for $p$. | $p$ | $t$ | $p \wedge t$ | $(p \wedge t) \rightarrow p$ | |-------|-------|---------------|-------------------------------| | T | T | T | T | | F | T | F | T |

3Step 3: Analyze the truth table

By looking at the truth table, we can see that the compound proposition $(p \wedge t) \rightarrow p$ has a truth value of True (T) in both cases, regardless of the truth value of $p$. Therefore, the truth value of $(p \wedge t) \rightarrow p$ is True (T).

Key Concepts

TautologyTruth TableLogical Proposition

Tautology

In logic, a tautology is an expression or statement that is always true, no matter what. This means that it holds true for every possible interpretation of its components. Tautologies are sometimes likened to mathematical truths that are axiomatic, as their truth is undeniable. For example, consider the expression $ p \lor eg p $. This is always true because either $ p $ is true, or it is not, making the expression valid under all circumstances.
Here are some key features of a tautology:

It does not depend on specific truths of individual propositions within it.
It is a fundamental building block in logic, helping to simplify complex expressions.
Tautologies are used to verify logical arguments and are a core part of logical reasoning.

The concept of tautology is crucial because it provides a baseline of truth that we can rely upon when constructing logical arguments or propositions. They serve as tools to check the validity and soundness of logical derivations.

Truth Table

A truth table is an essential tool in mathematics and logic that helps us systematically explore the truth values of logical expressions. It generally lists all the possible combinations of truth values for a given set of propositions and determines the corresponding truth value for each combination in the logical expression.
This is key to understanding how complex propositions behave under different conditions. For the expression $(p \wedge t) \rightarrow p$, we construct a table to determine if it's always true. The table might look like:

One row for each possible truth value of the propositions $p$ and $t$.
Columns representing $p \wedge t$ and $(p \wedge t) \rightarrow p$.

The truth table enables us to see:
- How each section of the proposition resolves when taking each possible truth value.
- Provides evidence of whether the overall proposition is a tautology or not.

Logical Proposition

A logical proposition is simply a statement in logic that can be either true or false. In logical terms, propositions are the building blocks used to create more complex logical structures through logical connectives, such as AND ($\wedge$), OR ($\lor$), and IMPLIES ($\rightarrow$).
Consider the following when working with logical propositions:

Each proposition corresponds to a specific truth value (true or false).
They allow the creation of complex logical expressions when combined with other propositions.
The overall truth of a complex proposition depends on both the individual truth values and the logical operators used to connect them.

In our original exercise, the logical proposition $(p \wedge t) \rightarrow p$ involves using the connective IMPLIES ($\rightarrow$). Even though one proposition may vary (like $p$), because $t$ is a tautology, parts of the truth table remain unaffected, allowing us to conclude the resulting proposition’s truth value with certainty.

Problem 50

Problem 51

Other exercises in this chapter

Problem 49

Let UD = set of integers, $P(x, y): x$ is a multiple of $y,$ and $Q(x, y): x \geq y$ Determine the truth value of each proposition. $$(\exists x)[\mathbf{

View solution

Problem 50

Simplify each boolean expression. $$p \vee(p \vee q)$$

View solution

Problem 51

Simplify each boolean expression. $$p \vee(\sim p \wedge q)$$

View solution

Problem 51

Let $t$ be a tautology and $p$ an arbitrary proposition. Find the truth value of each. $$p \rightarrow(p \wedge t)$$

View solution