Problem 19 - Thinking Mathematically

Question

Let \(p\) and \(q\) represent the following simple statements: \(p: Y o u\) are human. q: You have feathers. Write each compound statement in symbolic form. You do not have feathers if you are human.

Step-by-Step Solution

Answer

The symbolic form of the compound statement 'You do not have feathers if you are human' is \(p \rightarrow \sim q\).

Step 1: Identify the Simple Statements

The simple statements \(p\) and \(q\) are defined as follows: \(p\): you are human, and \(q\): you have feathers.

Step 2: Identify the Compound Statement

The compound statement is 'You do not have feathers if you are human.', which can be broken down into 'if you are human' (hypothesis) and 'you do not have feathers' (conclusion). The word 'if' indicates a conditional statement.

Step 3: Translate into Symbolic Form

The conditional statement can be translated into symbolic form as follows: 'if you are human' corresponds to \(p\), and 'you do not have feathers' corresponds to \(\sim q\) (where 'not' is represented by \(\sim\)). Thus, the compound statement 'You do not have feathers if you are human' would be represented symbolically as \(p \rightarrow \sim q\).

Key Concepts

Simple Statements

In symbolic logic, a simple statement is a basic assertion that doesn't contain any other statements within it. In our exercise, these are statements like "You are human" and "You have feathers." These statements can be true or false, represented by variables in symbolic form. Here, the symbol \( p \) corresponds to "You are human," and \( q \) corresponds to "You have feathers." Each of these statements is fundamental and stands alone without any logical operations or connectives.

Simple statements are the building blocks for more complex logical expressions. Understanding them is essential, as they form the basis upon which all compound statements are constructed.

Compound Statements

Compound statements combine two or more simple statements using logical connectives such as "and," "or," "if... then," and "not." In our exercise, the statement "You do not have feathers if you are human" is a compound statement. This involves not just the simple statements but also logical relations defined by the sentence structure.

To analyze and translate this into symbolic form, recognize that the simple statement "You have feathers" (\( q \)) is negated to "You do not have feathers" (\( \sim q \)). Such logical expressions are crucial in reasoning and are used for drawing connections between propositions.

The understanding of compound statements is vital in symbolic logic as it shows how different statements are interrelated, allowing complex ideas or conditions to be represented and manipulated logically.

Conditional Statement

A conditional statement uses the "if... then" structure to combine two propositions. It describes a relationship where one proposition (the hypothesis) implies another proposition (the conclusion). In our exercise, the statement "if you are human, then you do not have feathers" is a prime example.

Here, "if you are human" places \( p \) as the antecedent, and "then you do not have feathers" places \( \sim q \) as the consequent. The symbolic representation \( p \rightarrow \sim q \) captures this relationship, where the arrow "\( \rightarrow \)" reads as "implies."


Conditional statements are a fundamental aspect of logical reasoning, emphasizing causality or dependency between propositions. This concept is used widely in mathematics, computer science, and philosophy to explain and analyze conditional scenarios or hypotheses.