Truth Values Of Predicates: A Math Exploration

by TextBrain Team 47 views

Hey guys! Today, let's dive into a super interesting math problem involving predicates, quantifiers, and truth values. We're going to break down a quadratic equation and explore when it's true for some or all real numbers. Buckle up, it's gonna be a fun ride!

Understanding the Predicate p(x)

So, we've got this predicate p(x): x^2 - 2x + 3 = 0. What does that even mean? Well, a predicate is basically a statement that can be true or false depending on the value of x. In this case, x is a real number, meaning it can be any number on the number line – integers, fractions, decimals, you name it! Our mission is to figure out if there's even one real number that makes this equation true, and whether it's true for every real number.

First, let's analyze the quadratic equation x^2 - 2x + 3 = 0. We can try to solve it using the quadratic formula, which is -b ± √(b^2 - 4ac) / 2a. In our equation, a = 1, b = -2, and c = 3. Plugging these values into the formula, we get:

x = (2 ± √((-2)^2 - 4 * 1 * 3)) / (2 * 1) x = (2 ± √(4 - 12)) / 2 x = (2 ± √(-8)) / 2

Uh oh! We've got a negative number under the square root. This means the solutions for x are complex numbers, not real numbers. Since we're only interested in real number solutions, this is a crucial piece of information.

Therefore, there is no real number x that satisfies the equation x^2 - 2x + 3 = 0. This means for any real number you pick, x^2 - 2x + 3 will never equal zero. It will always be some other value. Keep this in mind as we proceed.

Evaluating (∃x) p(x): "There Exists an x such that p(x)"

The proposition (∃x) p(x) reads as "There exists an x such that p(x) is true." In simpler terms, it's asking: "Is there at least one real number x that makes the equation x^2 - 2x + 3 = 0 true?"

Given our analysis above, we already know the answer is no. There are no real solutions to the quadratic equation. Thus, the proposition (∃x) p(x) is false. To reiterate, we found that the discriminant (the part under the square root in the quadratic formula) is negative, indicating that the roots are complex, not real. Because we are specifically looking for real number solutions, the existence claim falls apart. Think of it like searching for a unicorn in your backyard – it just ain't gonna happen!

Evaluating (∀x) p(x): "For All x, p(x)"

Next, we have the proposition (∀x) p(x), which translates to "For all x, p(x) is true." In simpler terms: "Does every real number x make the equation x^2 - 2x + 3 = 0 true?"

Again, we know from our previous work that this is definitely not the case. We've already established that no real number satisfies the equation. If even one real number fails to make the equation true, then the statement "for all x" is automatically false. Therefore, the proposition (∀x) p(x) is also false. Imagine trying to convince everyone on Earth that the moon is made of cheese – it's just not true, and you won't succeed!

Verifying Negation Rules

Now, let's check out the negation rules. These rules tell us how to negate quantified statements. They're super handy for understanding the relationships between "exists" and "for all."

  • Negation of (∃x) p(x): The negation of "There exists an x such that p(x)" is "For all x, p(x) is not true." Symbolically, this is written as ¬(∃x) p(x) ≡ (∀x) ¬p(x). In our case, this means "For all real numbers x, x^2 - 2x + 3 ≠ 0."
  • Negation of (∀x) p(x): The negation of "For all x, p(x)" is "There exists an x such that p(x) is not true." Symbolically, this is ¬(∀x) p(x) ≡ (∃x) ¬p(x). In our context, this means "There exists a real number x such that x^2 - 2x + 3 ≠ 0."

Let’s verify these rules based on our previous findings:

  1. Negating (∃x) p(x): We found that (∃x) p(x) is false. Therefore, its negation, ¬(∃x) p(x), should be true. According to the rule, ¬(∃x) p(x) is equivalent to (∀x) ¬p(x), which means "For all real numbers x, x^2 - 2x + 3 ≠ 0." Is this true? Yes, it is! We determined earlier that no real number x satisfies the equation x^2 - 2x + 3 = 0. So, for every real number, the equation will not equal zero. The negation rule holds!

  2. Negating (∀x) p(x): We found that (∀x) p(x) is false. Its negation, ¬(∀x) p(x), should therefore be true. The rule tells us that ¬(∀x) p(x) is equivalent to (∃x) ¬p(x), which means "There exists a real number x such that x^2 - 2x + 3 ≠ 0." Is this true? Absolutely! Since no real number makes the equation equal to zero, then every real number makes the equation not equal to zero. So, there certainly exists at least one (in fact, infinitely many) real number that satisfies this condition. The negation rule checks out here too!

Conclusion

Alright, guys, we've successfully navigated this mathematical journey! We determined that for the predicate p(x): x^2 - 2x + 3 = 0, both (∃x) p(x) and (∀x) p(x) are false when x is a real number. We then meticulously verified the negation rules for these quantified statements, confirming their validity.

This exercise showcases the importance of understanding quantifiers and how they interact with predicates. It also highlights the significance of carefully analyzing equations to determine their solutions and truth values. Keep practicing these kinds of problems, and you'll become a math whiz in no time! Keep exploring and having fun with math! Peace out!