Unveiling The Truth Set Of (∃x ∈ N) (x+3<5) A Step-by-Step Guide

In the realm of mathematical propositions, understanding the truth set is crucial for grasping the meaning and validity of a statement. This article delves into the proposition (∃x ∈ N) (x+3<5), dissecting its components and systematically determining its truth set. Our journey will involve unraveling the symbols, exploring the domain of natural numbers, and solving the inequality to pinpoint the values of 'x' that make the proposition true. By the end, we will have a clear understanding of the proposition's truth set and the underlying mathematical principles at play.

Breaking Down the Proposition

To decipher the proposition (∃x ∈ N) (x+3<5), we need to understand the meaning of each symbol and component. Let's break it down:

• ∃: This symbol represents the existential quantifier, which means "there exists" or "for some." It asserts that there is at least one element that satisfies the condition that follows.
• x: This is a variable, representing an unknown number.
• ∈: This symbol means "is an element of" or "belongs to." It indicates that the variable 'x' belongs to a specific set.
• N: This denotes the set of natural numbers. Natural numbers are positive integers, typically starting from 1 (i.e., 1, 2, 3, ...). Some definitions include 0 as a natural number, but for this context, we'll consider natural numbers starting from 1.
• (x+3<5): This is the inequality that the variable 'x' must satisfy. It states that the sum of 'x' and 3 must be less than 5.

Putting it all together, the proposition (∃x ∈ N) (x+3<5) reads as: "There exists a natural number 'x' such that x+3 is less than 5."

This proposition is a statement about the existence of a natural number that fulfills a specific condition. To determine its truth set, we must find all natural numbers 'x' that make the inequality x+3<5 true. This involves solving the inequality and identifying the natural numbers within the solution set.

Exploring the Domain: Natural Numbers (N)

The domain of the variable 'x' is the set of natural numbers, denoted by N. Understanding the characteristics of natural numbers is essential for finding the truth set of the proposition. Natural numbers are the counting numbers, typically defined as positive integers starting from 1. They are whole numbers without any fractional or decimal parts.

• Natural Numbers: N = {1, 2, 3, 4, 5, ...}

It's important to note that the set of natural numbers is infinite, meaning it continues indefinitely. This characteristic plays a crucial role when determining the truth set of propositions, as we need to consider all possible natural number values.

In the context of our proposition (∃x ∈ N) (x+3<5), the domain of 'x' being natural numbers restricts the possible values that 'x' can take. We are only concerned with natural numbers that satisfy the inequality x+3<5. This limitation simplifies the process of finding the truth set, as we only need to test natural numbers.

Solving the Inequality: x+3<5

To find the truth set of the proposition, we need to solve the inequality x+3<5. This involves isolating the variable 'x' to determine the range of values that satisfy the inequality. Let's go through the steps:

1. Subtract 3 from both sides: x + 3 - 3 < 5 - 3 x < 2

This resulting inequality, x < 2, tells us that 'x' must be less than 2. However, we must remember that 'x' is restricted to the domain of natural numbers (N). This constraint significantly impacts the truth set.

Now, we need to identify the natural numbers that are less than 2. From the definition of natural numbers (N = {1, 2, 3, 4, 5, ...}), we can see that only one natural number satisfies this condition.

Identifying the Truth Set

Having solved the inequality x+3<5 and determined that x<2, we now focus on identifying the natural numbers that fall within this solution set. Recall that the set of natural numbers (N) consists of positive integers starting from 1: N = {1, 2, 3, 4, 5, ...}.

Given the inequality x<2, we seek natural numbers that are strictly less than 2. Examining the set of natural numbers, we find that only one number fits this criterion:

• x = 1

The number 1 is a natural number and is less than 2. The next natural number, 2, does not satisfy the condition x<2 because 2 is not strictly less than 2. Therefore, the truth set for the inequality x<2 within the domain of natural numbers contains only one element.

Expressing the Truth Set

The truth set is the set of all values of 'x' that make the proposition true. In this case, the proposition is (∃x ∈ N) (x+3<5), and we have determined that the only natural number that satisfies the inequality x+3<5 (which simplifies to x<2) is 1. Therefore, the truth set can be expressed as:

• Truth Set = {1}

This means that the proposition (∃x ∈ N) (x+3<5) is true because there exists at least one natural number (namely, 1) that satisfies the condition x+3<5. If we substitute x=1 into the original inequality, we get 1+3<5, which simplifies to 4<5, a true statement.

Analyzing the Given Options

Now that we have determined the truth set of the proposition (∃x ∈ N) (x+3<5) to be {1}, we can analyze the given options and identify the correct one. The options are:

• A {1}
• B {2}
• C {1, 2}
• D {1, 2, 3}
• E {1, 2, 3, 4}

Comparing our calculated truth set {1} with the given options, we can clearly see that option A matches our result.

• Option A: {1}

The truth set {1} indicates that the only natural number that satisfies the proposition (∃x ∈ N) (x+3<5) is 1. This aligns perfectly with our analysis and calculations. The other options include numbers that do not satisfy the inequality x+3<5, making them incorrect.

Conclusion: The Correct Answer

Based on our comprehensive analysis and step-by-step solution, we have determined that the truth set of the proposition (∃x ∈ N) (x+3<5) is {1}. This means that the only natural number that satisfies the condition x+3<5 is 1.

Therefore, the correct answer among the given options is:

• A {1}

This conclusion is reached by carefully breaking down the proposition, understanding the domain of natural numbers, solving the inequality, identifying the truth set, and comparing it with the provided options. The process demonstrates the importance of logical reasoning and mathematical principles in determining the validity of propositions.

By thoroughly analyzing the proposition and its components, we have not only found the correct answer but also gained a deeper understanding of mathematical propositions and truth sets. This knowledge is invaluable for further explorations in mathematics and logic.