Solving Math Predicates: A Step-by-Step Guide

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Solving Math Predicates: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of predicates and propositions. We'll be tackling some interesting problems involving unary predicates. Ready to flex those math muscles? Let's get started!

Understanding the Predicates: p(x) and q(y)

First things first, let's understand what we're dealing with. We've got two unary predicates: p(x) and q(y). A unary predicate is a statement about a single variable. Think of it like a function that, when you plug in a value (the variable), spits out a true or false answer.

The predicate p(x) is defined as: 2(1 - x) + 0.6(3x - 5) = -1.3, where x belongs to the set of real numbers (R). This means we can plug in any real number for x and see if the equation holds true. Basically, we are checking if the equation is true, if it is, the proposition is true, if it isn't the proposition is false. Let's start with this one in detail, so we get the main idea.

The predicate q(y) is defined as: (y - 3)^2 + (y + 4)^2 - (y - 5)^2 = 17y + 24, where y belongs to the set of natural numbers (N). Here, we can only plug in natural numbers (1, 2, 3, and so on) for y. Again, the goal is to see if the equation holds true for the specific value of y we input. We will follow the same logic as the previous explanation. Remember that the main goal of this is to determine if the proposition is true or false based on the value.

Breaking Down the Predicates

Let's break down the given predicates to get a better understanding of them. The predicate p(x) is a linear equation. This means it involves variables raised to the power of 1. When we look at this in a graphical form, this will form a line on a 2D plane. When solving p(x), the objective is to determine what value of x will satisfy the given equation. This can be done by isolating the variable x on one side of the equation and then simplifying it.

The predicate q(y) appears to be a quadratic equation initially, due to the squared terms. However, as we expand and simplify the expression, we might find that the squared terms cancel out, and the equation simplifies to a linear form. In q(y), we'll be dealing with natural numbers. Natural numbers are whole numbers, so the solution, or the value, has to be a whole number for the proposition to be considered true. This means that if we determine a value that is not an integer, we can automatically state that the proposition is false.

These initial insights into the structure of these predicates helps us to determine the right method to evaluate them. Now, let's evaluate them using the given specific values to have a clearer understanding.

Evaluating p(x): Particular Propositions

Now, let's get down to the nitty-gritty and evaluate the particular propositions for p(x). This involves substituting the given values of x into the equation and checking if the equation holds true. Specifically, we'll assess p(-1/2) and p(√3).

Calculating p(-1/2)

For p(-1/2), we substitute x = -1/2 into the equation 2(1 - x) + 0.6(3x - 5) = -1.3. Let's do it step by step:

  1. Substitute: 2(1 - (-1/2)) + 0.6(3(-1/2) - 5) = -1.3
  2. Simplify within parentheses: 2(1 + 1/2) + 0.6(-3/2 - 5) = -1.3
  3. Further simplification: 2(3/2) + 0.6(-13/2) = -1.3
  4. Multiply: 3 + (-3.9) = -1.3
  5. Calculate: -0.9 = -1.3

Since -0.9 does not equal -1.3, the proposition p(-1/2) is false.

Calculating p(√3)

Now, let's find the value for p(√3). We substitute x = √3 into the original equation: 2(1 - x) + 0.6(3x - 5) = -1.3.

  1. Substitute: 2(1 - √3) + 0.6(3√3 - 5) = -1.3
  2. Expand: 2 - 2√3 + 1.8√3 - 3 = -1.3
  3. Combine like terms: -1 - 0.2√3 = -1.3
  4. Isolate the radical: -0.2√3 = -0.3
  5. Solve for the radical: √3 = 1.5

As we know, the square root of 3 is approximately 1.732, and clearly is not equal to 1.5. Hence, the proposition p(√3) is also false.

Evaluating q(y): Particular Propositions

Let's switch gears and evaluate the propositions related to q(y). We will substitute the specified values of y into the equation. Remember that the domain for q(y) is the set of natural numbers, meaning that only positive whole numbers are valid substitutions.

Calculating q(-3)

We start with q(-3). Substitute y = -3 into the equation (y - 3)^2 + (y + 4)^2 - (y - 5)^2 = 17y + 24:

  1. Substitute: (-3 - 3)^2 + (-3 + 4)^2 - (-3 - 5)^2 = 17(-3) + 24
  2. Simplify within parentheses: (-6)^2 + (1)^2 - (-8)^2 = -51 + 24
  3. Calculate the powers: 36 + 1 - 64 = -27
  4. Simplify: -27 = -27

Since the equation holds true, the proposition q(-3) is true. However, because y belongs to the set of natural numbers, and -3 is not a natural number, then q(-3) is false.

Calculating q(0)

Now, let's evaluate q(0). Substitute y = 0 into the equation (y - 3)^2 + (y + 4)^2 - (y - 5)^2 = 17y + 24:

  1. Substitute: (0 - 3)^2 + (0 + 4)^2 - (0 - 5)^2 = 17(0) + 24
  2. Simplify within parentheses: (-3)^2 + (4)^2 - (-5)^2 = 0 + 24
  3. Calculate the powers: 9 + 16 - 25 = 24
  4. Simplify: 0 = 24

Since 0 does not equal 24, the proposition q(0) is false.

Final Thoughts and Key Takeaways

Alright, guys, we've walked through solving these math predicates step by step! Here's a quick recap of the important stuff:

  • Predicates: Propositions that are either true or false depending on the value of the variable.
  • Unary Predicates: Predicates involving a single variable.
  • Evaluating Propositions: Substituting values into the predicate and determining if the resulting statement is true or false.

We've learned how to substitute values into the equations and then simplify them. It's really all about applying the basic principles of algebra. This includes the order of operations, simplifying expressions, and solving equations. The key is to take it slow and steady, one step at a time.

I really hope this was helpful! Keep practicing, and you'll become a predicate pro in no time! Remember, the more you practice, the easier it gets. Feel free to try some similar problems on your own, and if you have any questions, don't hesitate to ask. Happy calculating, and keep exploring the amazing world of mathematics! Understanding these concepts is essential for a strong foundation in mathematics.