Mathematical Logic

Concept:
Use identity: Step 1: Use given.
Step 2: Simplify expression. Gives value = 3
Step 3: Conclusion. Answer = 3

Concept:
Use substitution method with trigonometric identity. Step 1: Rewrite numerator.
Step 2: Substitute. Let:
Step 3: Transform integral.
Step 4: Split integral.
Step 5: Back substitute.
Step 6: Conclusion. Correct option is (D).

Concept: In propositional logic:

• Tautology: A statement that is always true.
• Contradiction: A statement that is always false.
• Contingency: A statement that is sometimes true and sometimes false.

The implication is false only when is true and is false.
Step 1: {Construct the truth table.}
Step 2: {Analyze the final column.} The final column contains both and . Thus, the statement is sometimes true and sometimes false.

Step 1: Understanding the Question:
We need to determine if the given implications are tautologies or false statements by analyzing the truth conditions of the premises.

Step 2: Key Formula or Approach:
1. is false only when and .
2. is false when and have different truth values ( or ).
3. is false only when both and are false.

Step 3: Detailed Explanation:
Evaluating R:
Premise: is false. This implies and .
Conclusion: . If and , then .
Since the conclusion is True when the premise is True, the statement "then is false" is False. Thus, R is False.

Evaluating S:
Premise: is false. This occurs if ( ) or ( ).
Case 1: . Then .
Case 2: . Then .
In both cases where the premise is true, the conclusion is True. Thus, the statement "then is false" is False. Thus, S is False.

Step 4: Final Answer:
Both R and S are false statements.

Concept:
For a conditional statement : Also remember:

• Conjunction is True only when both are True.
• Disjunction is False only when both are False.

Step 1: {Use the condition for implication to be false.} Given: An implication is False only when:
Step 2: {Evaluate and .} For the conjunction: both statements must be True:
Step 3: {Evaluate and .} For the disjunction: both components must be False:
Step 4: {Write the final truth values.}
Step 5: {Conclusion.} Thus, the required truth values are: