All questions of Discrete Mathematics for Computer Science Engineering (CSE) Exam

S: We should be cautious.

This question is incomplete and cannot be answered. Please provide the full options.

Understanding Logical Equivalence
In propositional logic, two statements are logically equivalent if they have the same truth value in every possible scenario. Here, we will explore why "p ∧ q" is logically equivalent to "¬(p → ¬q".
Analyzing the Options
- Option A: ¬(p → ¬q)
- This expression can be rewritten using logical implications. The expression "p → ¬q" is equivalent to "¬p ∨ ¬q". Therefore, "¬(p → ¬q)" can be rephrased as "¬(¬p ∨ ¬q)".
- Applying De Morgan's laws, we can simplify this to "p ∧ q". Thus, "¬(p → ¬q)" is logically equivalent to "p ∧ q", confirming that Option A is correct.
- Option B: (p → ¬q)
- This suggests that if p is true, then q must be false, which does not reflect the conjunction of p and q.
- Option C: (¬p → ¬q)
- This means if p is false, then q must also be false, which is not equivalent to p and q being true simultaneously.
- Option D: (¬p → q)
- This indicates that if p is false, then q is true, which again does not capture the essence of "p ∧ q".
Conclusion
Based on the analysis above, the correct answer is indeed option A: "¬(p → ¬q)", as it is logically equivalent to "p ∧ q" using fundamental logical transformations and laws. This understanding is crucial for tackling problems in Computer Science Engineering and logic-based reasoning.

The statement "L(x, y)" is incomplete and does not provide enough information to determine its meaning. In order to understand the meaning of L(x, y), additional context or information is needed.

In order to determine which rule of inference is used in each argument, we need to see the specific arguments. Please provide the arguments you would like to analyze.

Explanation:

A proposition is a statement that is either true or false. It is a declarative sentence that can be evaluated as either being true or false. Let's analyze each option to determine if it is a proposition or not.

a) Get me a glass of milkshake:
This is not a proposition because it is a command or a request. It does not make a claim that can be evaluated as true or false.

b) God bless you!
This is not a proposition either. It is an expression of well-wishing or a blessing. It does not make a factual claim that can be evaluated as true or false.

c) What is the time now?
This is not a proposition because it is a question. It is seeking information rather than making a claim.

d) The only odd prime number is 2
This is a proposition because it makes a factual claim that can be evaluated as true or false. In this case, the statement is false because the number 2 is not an odd prime number; it is the only even prime number.

In conclusion, the correct answer is option D: The only odd prime number is 2 because it is the only statement that qualifies as a proposition.

Without the specific arguments provided, I am unable to determine which rule of inference is used in each argument. Could you please provide the arguments?

However, if you are referring to proofreading, it is recommended to proofread your work after finishing the first draft and before submitting it. This allows you to catch any errors or mistakes and make necessary corrections. It is also a good idea to take a break from your work before proofreading to gain a fresh perspective.

"the earth is flat" is false."

Explanation:

Understanding the statement:
The statement ∀x∃y(x < y)="" translates="" to="" "for="" all="" real="" numbers="" x,="" there="" exists="" a="" real="" number="" y="" such="" that="" x="" is="" less="" than="" />

Breaking down the statement:
- ∀x: This symbol (∀) represents "for all" or "for every." In this case, it indicates that the following statement applies to all real numbers x.
- ∃y: This symbol (∃) represents "there exists" or "there is." It indicates that there exists a real number y that satisfies the given condition.
- (x < y):="" this="" part="" of="" the="" statement="" states="" that="" x="" is="" less="" than="" />

Explanation of option 'A':
Option 'A' correctly translates the given statement by stating that for every real number x, there exists a real number y such that x is less than y. This interpretation accurately captures the meaning of the original statement in English.
Therefore, the correct answer is option 'A': "For all real number x there exists a real number y such that x is less than y."

Exhaustive Proof
An exhaustive proof is a type of proof that covers all possible cases. It is a method of proving a statement by considering and addressing each and every possible scenario or case.

Explanation
When dealing with a situation where there are multiple possibilities or cases, an exhaustive proof ensures that each and every case is considered and proven. This type of proof is particularly useful when the number of cases is finite and manageable.

In an exhaustive proof, the following steps are typically followed:
1. Identify all possible cases: The first step is to identify and list down all the possible cases or scenarios that need to be considered in order to prove the statement.
2. Address each case individually: For each case, the proof is carried out separately, providing evidence and reasoning specific to that particular case.
3. Cover all cases: It is important to ensure that all cases are covered and addressed in the proof. This means that no case should be left out or overlooked.
4. Present the proof: The proof is presented in a clear and organized manner, demonstrating the validity of the statement for each case.

Advantages of Exhaustive Proof
- Completeness: An exhaustive proof ensures that all possible cases are considered, providing a comprehensive and complete analysis of the statement.
- Confidence: By addressing each case individually, an exhaustive proof instills confidence in the validity of the statement, as it covers all possible scenarios.
- Accuracy: The thorough examination of all cases in an exhaustive proof minimizes the chances of errors or oversights.

Example
Let's consider a statement: "All prime numbers are odd." To prove this statement using an exhaustive proof, we would consider all possible cases:
1. Case 1: Prime number is 2 (the only even prime number). In this case, the statement is false, as 2 is not odd.
2. Case 2: Prime number is any other odd number. In this case, the statement is true, as all odd prime numbers are indeed odd.

By considering and addressing both cases, we have covered all possibilities and proven the statement. This is an example of an exhaustive proof.

Conclusion
Exhaustive proof is a method of proof that covers all possible cases. It is particularly useful when dealing with a finite number of cases and ensures the completeness and accuracy of the proof. By considering and addressing each case individually, an exhaustive proof provides confidence in the validity of the statement.

Unfortunately, you have not provided any information or example to determine what rule of inference is being used. Please provide more context or an example so that I can assist you further.

Understanding Logical Equivalences
In propositional logic, understanding how different logical expressions relate to each other is crucial. The expression "p ∨ q" (p or q) represents a disjunction, meaning at least one of p or q must be true.
Evaluating the Options
To determine which option is logically equivalent to p ∨ q, we can analyze each choice:
a) ¬q → ¬p
- This expression states that if q is false, then p must also be false.
- This does not represent the same truth values as p ∨ q.
b) q → p
- This means if q is true, then p is also true.
- Again, this does not capture the essence of p ∨ q.
c) ¬p → ¬q
- This implies if p is false, then q must also be false.
- This is not equivalent to p ∨ q.
d) ¬p → q
- This states that if p is false, then q must be true.
- This expression aligns with the truth table of p ∨ q. If p is false, q must be true for the disjunction to hold, making this the correct equivalent.
Truth Table Verification
Creating a truth table for p ∨ q and ¬p → q confirms:
- When both p and q are true: p ∨ q is true, ¬p → q is true.
- When p is true and q is false: p ∨ q is true, ¬p → q is true.
- When p is false and q is true: both p ∨ q and ¬p → q are true.
- When both p and q are false: p ∨ q is false, ¬p → q is false.
Hence, option d) ¬p → q is logically equivalent to p ∨ q.

The compound propositions p and q are called logically equivalent if the statement "p if and only if q" is a tautology.

Then q is "I love cricket."

Answer:

Given:
P: I am in Delhi.
Q: Delhi is clean.

To find:
q ^ p

Explanation:
The symbol ^ represents the logical operator "and". When two statements are connected by ^, both statements must be true for the combined statement to be true.

In this case, we have two statements: Q (Delhi is clean) and P (I am in Delhi). We want to find the combined statement q ^ p.

Step 1: Substitute the values of Q and P into the combined statement.
q ^ p = (Delhi is clean) ^ (I am in Delhi)

Step 2: Apply the logical operator ^ to the statements.
(Delhi is clean) ^ (I am in Delhi) = Delhi is clean and I am in Delhi.

Therefore, the correct answer is option 'A': Delhi is clean and I am in Delhi.