Truth Value Of Mathematical Statements: A Detailed Guide

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Hey guys! Today, we're diving into the fascinating world of mathematical statements and figuring out whether they're true or false. It might sound a bit intimidating, but trust me, it's super interesting once you get the hang of it. We'll be looking at some specific examples involving sets and intervals, so let's get started!

Understanding Mathematical Statements

Before we jump into the problems, let's quickly recap what we mean by a mathematical statement. Essentially, a mathematical statement is a declarative sentence that can be either true or false, but not both. Think of it like a light switch – it's either on (true) or off (false).

Key Concepts

  • ∈ (belongs to): This symbol means that an element is a member of a set. For example, 2 ∈ {1, 2, 3} means that 2 is an element of the set {1, 2, 3}. This seems straightforward, but it's the bedrock upon which we will evaluate the statements in question. This simple symbol of belonging is crucial in many areas of mathematics, including set theory and analysis.
  • ∉ (does not belong to): Conversely, this means that an element is not a member of a set. So, 4 ∉ {1, 2, 3} because 4 is not in the set.
  • ⊂ (subset of): This indicates that all elements of one set are also elements of another set. For instance, {1, 2} ⊂ {1, 2, 3} because every element in {1, 2} is also in {1, 2, 3}. This relationship is important in establishing hierarchies and connections between sets. Knowing that a set is a subset of another allows us to make inferences about the properties of elements within those sets.
  • Intervals: Intervals are continuous sets of numbers within specific boundaries. We use different notations to indicate whether the boundaries are included or excluded. Let's break it down:
    • [a, b]: This is a closed interval, meaning it includes both a and b. For example, [1, 5] includes all numbers between 1 and 5, as well as 1 and 5 themselves. Closed intervals are key in real analysis and are used extensively when dealing with continuous functions and their limits.
    • (a, b): This is an open interval, excluding both a and b. So, (1, 5) includes all numbers between 1 and 5, but not 1 and 5. Open intervals are crucial in defining concepts such as continuity and differentiability in calculus.
    • [a, b): This is a half-open interval, including a but excluding b. For example, [1, 5) includes 1 but not 5.
    • (a, b]: Another half-open interval, excluding a but including b. For instance, (1, 5] includes 5 but not 1.

Understanding these symbols and notations is absolutely crucial for tackling the questions we have ahead. Think of them as the alphabet of mathematical logic – you need to know them to read and write mathematical statements fluently!

Analyzing the Statements

Now, let's dive into the statements themselves and figure out their truth values. We'll go through each one step by step, so you can see the reasoning behind the answers.

a) -2.5 ∈ [-3; 0)

Our main keyword here is whether -2.5 belongs to the interval [-3, 0). Remember, [-3, 0) means all numbers from -3 (inclusive) up to 0 (exclusive). Think of it as a number line stretching from -3 to just before 0.

  • The Question: Is -2.5 within this range?
  • The Analysis: Well, -2.5 is greater than -3 and less than 0. So, it does indeed fall within the specified interval. In fact, to rigorously ascertain this, consider that -3 ≤ -2.5 < 0, which confirms that -2.5 lies within the interval [-3, 0). This type of precise verification is crucial in mathematical proofs and problem-solving.
  • The Verdict: Therefore, the statement -2.5 ∈ [-3; 0) is TRUE.

b) 4/5 ∈ [1; 2]

In this statement, we're checking if the fraction 4/5 belongs to the closed interval [1, 2]. This interval includes all numbers between 1 and 2, including 1 and 2 themselves.

  • The Question: Is 4/5 (which is 0.8) between 1 and 2?
  • The Analysis: 4/5 is equal to 0.8. Now, 0.8 is clearly less than 1. So, it's not within the interval [1, 2]. To be more specific, the interval [1, 2] contains all real numbers x such that 1 ≤ x ≤ 2. Since 0.8 < 1, it does not satisfy the condition for belonging to this interval. This demonstrates the importance of understanding number comparisons and interval definitions.
  • The Verdict: Hence, the statement 4/5 ∈ [1; 2] is FALSE.

c) 2^(1/2) ∉ (1; 3)

This one's a bit trickier. We're asking if the square root of 2 (2^(1/2)) does not belong to the open interval (1, 3). Remember, (1, 3) includes all numbers between 1 and 3, but excludes 1 and 3.

  • The Question: Is the square root of 2 not between 1 and 3?
  • The Analysis: The square root of 2 is approximately 1.414. This value is greater than 1 and less than 3. So, it does belong to the interval (1, 3). This situation highlights the importance of knowing common irrational numbers and their approximate values. Recognizing that √2 ≈ 1.414 allows us to quickly determine its place on the number line and within intervals.
  • The Verdict: Because the square root of 2 does belong to (1, 3), the statement 2^(1/2) ∉ (1; 3) is FALSE. It's a double negative, so you have to be extra careful!

d) [-3; 2) ⊂ [-4; 3]

Now we're looking at subsets. The question here is whether the interval [-3, 2) is a subset of the interval [-4, 3]. This means we need to check if every number in [-3, 2) is also in [-4, 3].

  • The Question: Is the entire interval [-3, 2) contained within the interval [-4, 3]?
  • The Analysis: The interval [-3, 2) includes all numbers from -3 (inclusive) up to 2 (exclusive). The interval [-4, 3] includes all numbers from -4 (inclusive) up to 3 (inclusive). Visualizing these intervals on a number line can be very helpful. You'll see that every number in [-3, 2) is also in [-4, 3]. This is because -4 ≤ -3 and 2 < 3. Therefore, the interval [-3, 2) lies entirely within [-4, 3].
  • The Verdict: So, the statement [-3; 2) ⊂ [-4; 3] is TRUE.

e) (-2; 0; 1} ⊂ [-2; 2]

This statement asks if the set {-2, 0, 1} is a subset of the interval [-2, 2]. Remember, {-2, 0, 1} is a set containing three specific numbers, while [-2, 2] is an interval containing all numbers between -2 and 2, including -2 and 2.

  • The Question: Are all the elements in the set {-2, 0, 1} also within the interval [-2, 2]?
  • The Analysis: We need to check each element individually:
    • -2: Is -2 in [-2, 2]? Yes, it is, because the interval includes -2.
    • 0: Is 0 in [-2, 2]? Yes, it's between -2 and 2.
    • 1: Is 1 in [-2, 2]? Yes, it's also between -2 and 2. Since all the elements in the set are within the interval, the set is indeed a subset of the interval. To be extremely precise, the interval [-2, 2] is the set of all real numbers x such that -2 ≤ x ≤ 2. Since -2, 0, and 1 all satisfy this condition, the set {-2, 0, 1} is a subset of [-2, 2].
  • The Verdict: Thus, the statement {-2; 0; 1} ⊂ [-2; 2] is TRUE.

Final Thoughts

So there you have it! We've gone through each statement, analyzed it, and determined its truth value. I hope this breakdown has helped you understand how to approach these types of problems. Remember, the key is to understand the definitions and symbols, and then carefully apply them to each situation. Keep practicing, and you'll become a pro at evaluating mathematical statements in no time! Understanding these concepts is vital not only for academic success but also for developing critical thinking skills that are valuable in all aspects of life.

If you have any more questions or want to explore other mathematical concepts, just let me know. Happy learning!