Predicates On A Set: Forming And Determining Truth Sets

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Predicates on a Set: Forming and Determining Truth Sets

Hey guys! Let's dive into a fascinating topic in mathematics: predicates on a set. Specifically, we're going to explore how to form new predicates from existing ones and how to determine their truth sets. We will take an in-depth look at predicates, truth sets, logical operations applied to predicates, and how to methodically solve problems involving these concepts.

Understanding Predicates and Truth Sets

Before we jump into the problem, let's make sure we're all on the same page with the basics. A predicate, in mathematical logic, is a statement that can be either true or false depending on the values of its variables. Think of it as a sentence with blanks that you can fill in. For example, x > 5 is a predicate. If x is 10, the predicate is true; if x is 2, it's false. This flexibility based on variable values is what makes predicates powerful tools in mathematical reasoning.

Now, the truth set of a predicate is the set of all values for the variables that make the predicate true. It's like the solution set for an equation, but for a logical statement. So, for the predicate x > 5, the truth set would be all numbers greater than 5. Understanding this, we can see how predicates help us define conditions and filter elements within a set.

Binary Predicates: A Closer Look

In this particular problem, we're dealing with binary predicates, which means predicates with two variables, typically denoted as p(x, y). These predicates describe a relationship between two elements. Our goal is to manipulate and combine these predicates using logical operations. So, the predicate p(x, y): x - 2y = 3 is a binary predicate because its truth depends on the values of both x and y. We'll explore how to determine when such predicates are true or false within a specified set.

The Problem at Hand

Okay, let's break down the problem we've got. We're given a set A = {1, 2, 3, 4, 5} and two binary predicates defined on this set:

  • p(x, y): x - 2y = 3
  • q(x, y): x + 3y = 8

Our task has two parts:

a) We need to form two new predicates using logical operations:

  • p(x, y) OR q(x, y) (the disjunction)
  • p(x, y) AND q(x, y) (the conjunction)

b) We need to determine the truth set for each of these newly formed predicates. That is, we need to find all pairs (x, y) from set A that make each predicate true.

This involves understanding how logical 'OR' and 'AND' operations work and applying them within the context of predicates. The truth set of a combined predicate relies on the individual truth values of its components.

Part a: Forming New Predicates

Understanding Logical Operations

Before we dive into forming the new predicates, let's quickly recap the logical operations OR (∨) and AND (∧). These are fundamental to mathematical logic and crucial for combining predicates.

  • The OR operation (also known as disjunction) is true if at least one of the predicates is true. In other words, p OR q is true if p is true, or q is true, or both are true. It's only false if both p and q are false.
  • The AND operation (also known as conjunction) is true only if both predicates are true. That is, p AND q is true if and only if both p and q are true. If either p or q (or both) are false, then p AND q is false.

Applying the Operations to Our Predicates

Now, let's apply these operations to our predicates p(x, y) and q(x, y):

  1. p(x, y) OR q(x, y): This predicate will be true if either x - 2y = 3 is true, or x + 3y = 8 is true, or both are true. We're looking for pairs (x, y) that satisfy at least one of these equations.
  2. p(x, y) AND q(x, y): This predicate will be true only if both x - 2y = 3 and x + 3y = 8 are true. We're looking for pairs (x, y) that simultaneously satisfy both equations.

By understanding the nature of these logical operations, we can systematically approach finding the truth sets for these combined predicates. The next step is to apply these definitions to the set A and identify the pairs that fulfill our conditions.

Part b: Determining the Truth Sets

Methodical Approach

To find the truth sets, we need to systematically check all possible pairs (x, y) from the set A = {1, 2, 3, 4, 5}. We'll plug each pair into our predicates and see if they hold true.

It might seem tedious, but a methodical approach ensures we don't miss any solutions. We'll create a table to keep track of our findings. This helps in visualizing and organizing the results.

Truth Set for p(x, y) OR q(x, y)

Let's start with p(x, y) OR q(x, y). We need to find pairs (x, y) that satisfy either x - 2y = 3 or x + 3y = 8 (or both).

We will test every pair (x, y) from the Cartesian product A x A. Remember, the Cartesian product is the set of all possible ordered pairs where the first element comes from the first set and the second element comes from the second set. In our case, since both sets are A, it means we consider all pairs where both x and y can be any number from 1 to 5.

Step-by-step Testing:

  1. (1, 1):
    • p(1, 1): 1 - 2(1) = -1 ≠ 3 (False)
    • q(1, 1): 1 + 3(1) = 4 ≠ 8 (False)
    • p(1, 1) OR q(1, 1): False
  2. (1, 2):
    • p(1, 2): 1 - 2(2) = -3 ≠ 3 (False)
    • q(1, 2): 1 + 3(2) = 7 ≠ 8 (False)
    • p(1, 2) OR q(1, 2): False
  3. (1, 3):
    • p(1, 3): 1 - 2(3) = -5 ≠ 3 (False)
    • q(1, 3): 1 + 3(3) = 10 ≠ 8 (False)
    • p(1, 3) OR q(1, 3): False
  4. (1, 4):
    • p(1, 4): 1 - 2(4) = -7 ≠ 3 (False)
    • q(1, 4): 1 + 3(4) = 13 ≠ 8 (False)
    • p(1, 4) OR q(1, 4): False
  5. (1, 5):
    • p(1, 5): 1 - 2(5) = -9 ≠ 3 (False)
    • q(1, 5): 1 + 3(5) = 16 ≠ 8 (False)
    • p(1, 5) OR q(1, 5): False
  6. (2, 1):
    • p(2, 1): 2 - 2(1) = 0 ≠ 3 (False)
    • q(2, 1): 2 + 3(1) = 5 ≠ 8 (False)
    • p(2, 1) OR q(2, 1): False
  7. (2, 2):
    • p(2, 2): 2 - 2(2) = -2 ≠ 3 (False)
    • q(2, 2): 2 + 3(2) = 8 (True)
    • p(2, 2) OR q(2, 2): True
  8. (2, 3):
    • p(2, 3): 2 - 2(3) = -4 ≠ 3 (False)
    • q(2, 3): 2 + 3(3) = 11 ≠ 8 (False)
    • p(2, 3) OR q(2, 3): False
  9. (2, 4):
    • p(2, 4): 2 - 2(4) = -6 ≠ 3 (False)
    • q(2, 4): 2 + 3(4) = 14 ≠ 8 (False)
    • p(2, 4) OR q(2, 4): False
  10. (2, 5):
    • p(2, 5): 2 - 2(5) = -8 ≠ 3 (False)
    • q(2, 5): 2 + 3(5) = 17 ≠ 8 (False)
    • p(2, 5) OR q(2, 5): False
  11. (3, 1):
    • p(3, 1): 3 - 2(1) = 1 ≠ 3 (False)
    • q(3, 1): 3 + 3(1) = 6 ≠ 8 (False)
    • p(3, 1) OR q(3, 1): False
  12. (3, 2):
    • p(3, 2): 3 - 2(2) = -1 ≠ 3 (False)
    • q(3, 2): 3 + 3(2) = 9 ≠ 8 (False)
    • p(3, 2) OR q(3, 2): False
  13. (3, 3):
    • p(3, 3): 3 - 2(3) = -3 ≠ 3 (False)
    • q(3, 3): 3 + 3(3) = 12 ≠ 8 (False)
    • p(3, 3) OR q(3, 3): False
  14. (3, 4):
    • p(3, 4): 3 - 2(4) = -5 ≠ 3 (False)
    • q(3, 4): 3 + 3(4) = 15 ≠ 8 (False)
    • p(3, 4) OR q(3, 4): False
  15. (3, 5):
    • p(3, 5): 3 - 2(5) = -7 ≠ 3 (False)
    • q(3, 5): 3 + 3(5) = 18 ≠ 8 (False)
    • p(3, 5) OR q(3, 5): False
  16. (4, 1):
    • p(4, 1): 4 - 2(1) = 2 ≠ 3 (False)
    • q(4, 1): 4 + 3(1) = 7 ≠ 8 (False)
    • p(4, 1) OR q(4, 1): False
  17. (4, 2):
    • p(4, 2): 4 - 2(2) = 0 ≠ 3 (False)
    • q(4, 2): 4 + 3(2) = 10 ≠ 8 (False)
    • p(4, 2) OR q(4, 2): False
  18. (4, 3):
    • p(4, 3): 4 - 2(3) = -2 ≠ 3 (False)
    • q(4, 3): 4 + 3(3) = 13 ≠ 8 (False)
    • p(4, 3) OR q(4, 3): False
  19. (4, 4):
    • p(4, 4): 4 - 2(4) = -4 ≠ 3 (False)
    • q(4, 4): 4 + 3(4) = 16 ≠ 8 (False)
    • p(4, 4) OR q(4, 4): False
  20. (4, 5):
    • p(4, 5): 4 - 2(5) = -6 ≠ 3 (False)
    • q(4, 5): 4 + 3(5) = 19 ≠ 8 (False)
    • p(4, 5) OR q(4, 5): False
  21. (5, 1):
    • p(5, 1): 5 - 2(1) = 3 (True)
    • q(5, 1): 5 + 3(1) = 8 (True)
    • p(5, 1) OR q(5, 1): True
  22. (5, 2):
    • p(5, 2): 5 - 2(2) = 1 ≠ 3 (False)
    • q(5, 2): 5 + 3(2) = 11 ≠ 8 (False)
    • p(5, 2) OR q(5, 2): False
  23. (5, 3):
    • p(5, 3): 5 - 2(3) = -1 ≠ 3 (False)
    • q(5, 3): 5 + 3(3) = 14 ≠ 8 (False)
    • p(5, 3) OR q(5, 3): False
  24. (5, 4):
    • p(5, 4): 5 - 2(4) = -3 ≠ 3 (False)
    • q(5, 4): 5 + 3(4) = 17 ≠ 8 (False)
    • p(5, 4) OR q(5, 4): False
  25. (5, 5):
    • p(5, 5): 5 - 2(5) = -5 ≠ 3 (False)
    • q(5, 5): 5 + 3(5) = 20 ≠ 8 (False)
    • p(5, 5) OR q(5, 5): False

The Truth Set:

After testing all pairs, we find that p(x, y) OR q(x, y) is true for (2, 2) and (5, 1). Therefore, the truth set for p(x, y) OR q(x, y) is {(2, 2), (5, 1)}.

Truth Set for p(x, y) AND q(x, y)

Now, let's find the truth set for p(x, y) AND q(x, y). This predicate is true only if both x - 2y = 3 and x + 3y = 8 are true.

We can refer back to our previous work where we tested all pairs for the OR condition. This time, we're only interested in the pairs where both p(x, y) and q(x, y) are true simultaneously.

Looking at our previous analysis, we see that only the pair (5, 1) satisfies both conditions:

  • p(5, 1): 5 - 2(1) = 3 (True)
  • q(5, 1): 5 + 3(1) = 8 (True)

The Truth Set:

Therefore, the truth set for p(x, y) AND q(x, y) is {(5, 1)}.

Conclusion

So, there you have it! We've successfully formed new predicates using logical operations and determined their truth sets. By methodically testing each pair and understanding the definitions of OR and AND, we were able to solve this problem. Remember, the key is to break down the problem into smaller steps and apply the definitions carefully. Keep practicing, and you'll become a predicate pro in no time!