What is the inference rule for the following logic?

(((p \to q) \wedge (q \to r)) \to (p \to r)) \equiv T

What is the inference rule for the following logic?

((p \wedge (p \to q)) \to q) \equiv T

Given the following valid argument, choose the best answer.

1. p \wedge (p \to q)

2. p

3. p \to q

4. q

Which inference rule is used to derive step 4?

In the class, the professor emphasized the importance of writing a proof with all statements step-by-step. Please visit chat.deepseek.com and ask this AI a question about the main idea of the Chain of Thought (CoT) and the relationship between CoT and writing a proof with all statements step-by-step. When asking this question, please click "DeepThink (R1)" so that the model can return its "thought" before the answer.

For this problem, you need to 1) copy and paste your question and the answer you got from DeepSeek R1, and 2) take a screenshot and upload it to demonstrate the "thought" of DeepSeek before answering your question.

In the class, the professor introduced three reasoning models: forward reasoning, backward reasoning, and the Chain of Thought (CoT). Please visit chat.deepseek.com and ask this AI a question about how forward reasoning and backward reasoning are related to the chain of thought (CoT). When asking this question, please click "DeepThink (R1)" so that the model can return its "thought" before the answer.

For this problem, you need to 1) copy and paste your question and the answer you got from DeepSeek R1, and 2) take a screenshot and upload it to demonstrate the "thought" of DeepSeek before answering your question.

Let x and y be integers.

Find the quantified predicate logic expression for the following statement.

The sum of two positive integers is positive.

Let x and y be integers.

Find the quantified predicate logic expression for the following statement.

The product of a positive integer and a negative integer is negative.

Let P(x , y) be the statement that x + y^2 = 3xy + 9, where x and y are integers.

Determine the Truth value of the following statement.

\neg \forall y (\neg (\exists x P(x, y)))

Let P(x , y) be the statement that x + y^2 = 3xy + 9, where x and y are integers.

Determine the Truth value of the following statement.

P(2, -1)

Let P(x , y) be the statement that x + y^2 = 3xy + 9, where x and y are integers.

Determine the Truth value of the following statement.

\forall y \exists x P(x, y)