Exercises1. In Each Of The Following, Two Open Statements \[$ P(x, Y) \$\] And \[$ Q(x, Y) \$\] Are Given, Where The Domain Of Both \[$ X \$\] And \[$ Y \$\] Is \[$ \mathbb{Z} \$\]. Determine The Truth Value Of

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Exercises in Mathematical Logic: Evaluating Truth Values of Statements

In the realm of mathematical logic, statements are used to express mathematical truths or falsehoods. These statements can be combined using logical operators to form more complex statements. In this article, we will explore exercises that involve evaluating the truth values of statements involving two open statements, P(x, y) and Q(x, y), where the domain of both x and y is the set of integers, ℤ.

Exercise 1: Evaluating Truth Values

Statement 1: P(x, y) and Q(x, y)

Two open statements, P(x, y) and Q(x, y), are given. We need to determine the truth value of the following statement:

P(x, y) ∧ Q(x, y)

where ∧ represents the logical conjunction operator.

Solution

To evaluate the truth value of P(x, y) ∧ Q(x, y), we need to consider the possible values of x and y. Since the domain of both x and y is ℤ, we can consider the following cases:

  • x = 0, y = 0
  • x = 0, y ≠ 0
  • x ≠ 0, y = 0
  • x ≠ 0, y ≠ 0

For each case, we need to determine the truth value of P(x, y) and Q(x, y). Let's assume that P(x, y) is true when x = y and false otherwise, and Q(x, y) is true when x ≠ y and false otherwise.

  • Case 1: x = 0, y = 0 P(x, y) is false, and Q(x, y) is false. Therefore, P(x, y) ∧ Q(x, y) is false.
  • Case 2: x = 0, y ≠ 0 P(x, y) is false, and Q(x, y) is true. Therefore, P(x, y) ∧ Q(x, y) is false.
  • Case 3: x ≠ 0, y = 0 P(x, y) is true, and Q(x, y) is false. Therefore, P(x, y) ∧ Q(x, y) is false.
  • Case 4: x ≠ 0, y ≠ 0 P(x, y) is true, and Q(x, y) is true. Therefore, P(x, y) ∧ Q(x, y) is true.

From these cases, we can see that the truth value of P(x, y) ∧ Q(x, y) depends on the values of x and y. Therefore, we cannot determine a fixed truth value for this statement.

Conclusion

In this exercise, we evaluated the truth value of the statement P(x, y) ∧ Q(x, y), where P(x, y) and Q(x, y) are open statements with a domain of ℤ. We considered four cases and determined the truth value of the statement for each case. The truth value of the statement depends on the values of x and y, and we cannot determine a fixed truth value for this statement.

Exercise 2: Evaluating Truth Values

Statement 2: P(x, y) and Q(x, y)

Two open statements, P(x, y) and Q(x, y), are given. We need to determine the truth value of the following statement:

P(x, y) ∨ Q(x, y)

where ∨ represents the logical disjunction operator.

Solution

To evaluate the truth value of P(x, y) ∨ Q(x, y), we need to consider the possible values of x and y. Since the domain of both x and y is ℤ, we can consider the following cases:

  • x = 0, y = 0
  • x = 0, y ≠ 0
  • x ≠ 0, y = 0
  • x ≠ 0, y ≠ 0

For each case, we need to determine the truth value of P(x, y) and Q(x, y). Let's assume that P(x, y) is true when x = y and false otherwise, and Q(x, y) is true when x ≠ y and false otherwise.

  • Case 1: x = 0, y = 0 P(x, y) is false, and Q(x, y) is false. Therefore, P(x, y) ∨ Q(x, y) is false.
  • Case 2: x = 0, y ≠ 0 P(x, y) is false, and Q(x, y) is true. Therefore, P(x, y) ∨ Q(x, y) is true.
  • Case 3: x ≠ 0, y = 0 P(x, y) is true, and Q(x, y) is false. Therefore, P(x, y) ∨ Q(x, y) is true.
  • Case 4: x ≠ 0, y ≠ 0 P(x, y) is true, and Q(x, y) is true. Therefore, P(x, y) ∨ Q(x, y) is true.

From these cases, we can see that the truth value of P(x, y) ∨ Q(x, y) depends on the values of x and y. However, we can see that the statement is true for all cases except one.

Conclusion

In this exercise, we evaluated the truth value of the statement P(x, y) ∨ Q(x, y), where P(x, y) and Q(x, y) are open statements with a domain of ℤ. We considered four cases and determined the truth value of the statement for each case. The truth value of the statement depends on the values of x and y, but we can see that the statement is true for all cases except one.

Exercise 3: Evaluating Truth Values

Statement 3: P(x, y) and Q(x, y)

Two open statements, P(x, y) and Q(x, y), are given. We need to determine the truth value of the following statement:

¬P(x, y)

where ¬ represents the logical negation operator.

Solution

To evaluate the truth value of ¬P(x, y), we need to consider the possible values of x and y. Since the domain of both x and y is ℤ, we can consider the following cases:

  • x = 0, y = 0
  • x = 0, y ≠ 0
  • x ≠ 0, y = 0
  • x ≠ 0, y ≠ 0

For each case, we need to determine the truth value of P(x, y). Let's assume that P(x, y) is true when x = y and false otherwise.

  • Case 1: x = 0, y = 0 P(x, y) is false. Therefore, ¬P(x, y) is true.
  • Case 2: x = 0, y ≠ 0 P(x, y) is false. Therefore, ¬P(x, y) is true.
  • Case 3: x ≠ 0, y = 0 P(x, y) is true. Therefore, ¬P(x, y) is false.
  • Case 4: x ≠ 0, y ≠ 0 P(x, y) is true. Therefore, ¬P(x, y) is false.

From these cases, we can see that the truth value of ¬P(x, y) depends on the values of x and y.

Conclusion

In this exercise, we evaluated the truth value of the statement ¬P(x, y), where P(x, y) is an open statement with a domain of ℤ. We considered four cases and determined the truth value of the statement for each case. The truth value of the statement depends on the values of x and y.

In this article, we evaluated the truth values of statements involving two open statements, P(x, y) and Q(x, y), where the domain of both x and y is ℤ. We considered three exercises and determined the truth value of the statement for each case. The truth value of the statement depends on the values of x and y, and we cannot determine a fixed truth value for these statements.
Q&A: Evaluating Truth Values of Statements

In our previous article, we explored exercises that involved evaluating the truth values of statements involving two open statements, P(x, y) and Q(x, y), where the domain of both x and y is the set of integers, ℤ. In this article, we will answer some frequently asked questions (FAQs) related to evaluating truth values of statements.

Q: What is the difference between a statement and a proposition?

A: A statement is a sentence that is either true or false, while a proposition is a statement that is always true or always false. In other words, a proposition is a statement that is not dependent on any variables or conditions.

Q: How do I determine the truth value of a statement?

A: To determine the truth value of a statement, you need to consider the possible values of the variables involved in the statement. You can use a truth table or a logical diagram to help you evaluate the statement.

Q: What is a truth table?

A: A truth table is a table that shows the truth values of a statement for all possible combinations of values of the variables involved in the statement. It is a useful tool for evaluating the truth value of a statement.

Q: What is a logical diagram?

A: A logical diagram is a visual representation of a statement that shows the relationships between the variables involved in the statement. It is a useful tool for evaluating the truth value of a statement.

Q: How do I evaluate the truth value of a statement involving logical operators?

A: To evaluate the truth value of a statement involving logical operators, you need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions inside the parentheses.
  2. Evaluate the expressions with the highest precedence (e.g., NOT, AND, OR).
  3. Evaluate the expressions with the next highest precedence (e.g., AND, OR).
  4. Evaluate the final expression.

Q: What is the difference between a conjunction and a disjunction?

A: A conjunction is a statement that is true only if both of the statements involved in the conjunction are true. A disjunction is a statement that is true if at least one of the statements involved in the disjunction is true.

Q: How do I evaluate the truth value of a statement involving negation?

A: To evaluate the truth value of a statement involving negation, you need to follow the rule:

¬P(x, y) is true if and only if P(x, y) is false.

Q: What is the difference between a statement and a formula?

A: A statement is a sentence that is either true or false, while a formula is a mathematical expression that can be true or false. A formula can involve variables, constants, and mathematical operations.

Q: How do I evaluate the truth value of a statement involving quantifiers?

A: To evaluate the truth value of a statement involving quantifiers, you need to follow the rule:

∀x P(x) is true if and only if P(x) is true for all values of x. ∃x P(x) is true if and only if P(x) is true for at least one value of x.

In this article, we answered some frequently asked questions (FAQs) related to evaluating truth values of statements. We hope that this article has been helpful in clarifying some of the concepts involved in evaluating truth values of statements. If you have any further questions, please don't hesitate to ask.

  • Statement: A sentence that is either true or false.
  • Proposition: A statement that is always true or always false.
  • Truth table: A table that shows the truth values of a statement for all possible combinations of values of the variables involved in the statement.
  • Logical diagram: A visual representation of a statement that shows the relationships between the variables involved in the statement.
  • Conjunction: A statement that is true only if both of the statements involved in the conjunction are true.
  • Disjunction: A statement that is true if at least one of the statements involved in the disjunction is true.
  • Negation: A statement that is true if and only if the statement being negated is false.
  • Quantifier: A symbol that indicates the scope of a statement, such as ∀ (for all) or ∃ (there exists).
  • Formula: A mathematical expression that can be true or false.