Civil Engineering (CE) Exam > Civil Engineering (CE) Questions > Let p, q, r, s represents the following propo...
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Let p, q, r, s represents the following propositions.
p: x ∈ {8, 9, 10, 11, 12}
q: x is a composite number
r: x is a perfect square
s: x is a prime number
The integer x ≥ 2 which satisfies ¬ ((p ⇒ q) ∧ (¬ r ∨ ¬ s)) is __________
p: x ∈ {8, 9, 10, 11, 12}
q: x is a composite number
r: x is a perfect square
s: x is a prime number
The integer x ≥ 2 which satisfies ¬ ((p ⇒ q) ∧ (¬ r ∨ ¬ s)) is __________
Correct answer is '11'. Can you explain this answer?
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Most Upvoted Answer
Let p, q, r, s represents the following propositions.p: x ∈ {8, 9...
Understanding the Propositions
To solve the expression ¬((p → q) ∧ (¬r ∨ ¬s), let's first clarify the propositions:
- p: x ∈ {8, 9, 10, 11, 12}
- q: x is a composite number
- r: x is a perfect square
- s: x is a prime number
Evaluating Each Proposition for x = 11
- For x = 11:
- p: True (11 is in the set)
- q: False (11 is prime, not composite)
- r: False (11 is not a perfect square)
- s: True (11 is prime)
Breaking Down the Expression
1. Evaluate p → q:
- This translates to "If p is true, then q is true."
- Since p is true and q is false, p → q is false.
2. Evaluate ¬r:
- Since r is false, ¬r is true.
3. Evaluate ¬s:
- Since s is true, ¬s is false.
4. Evaluate ¬r ∨ ¬s:
- This is true (true ∨ false = true).
5. Combine Results:
- Now substituting back: ¬((p → q) ∧ (¬r ∨ ¬s)) becomes ¬(false ∧ true).
- This simplifies to ¬(false), which is true.
Conclusion
Since the expression is true for x = 11, it satisfies the condition. Hence, the integer x ≥ 2 that satisfies the original expression is:
11
To solve the expression ¬((p → q) ∧ (¬r ∨ ¬s), let's first clarify the propositions:
- p: x ∈ {8, 9, 10, 11, 12}
- q: x is a composite number
- r: x is a perfect square
- s: x is a prime number
Evaluating Each Proposition for x = 11
- For x = 11:
- p: True (11 is in the set)
- q: False (11 is prime, not composite)
- r: False (11 is not a perfect square)
- s: True (11 is prime)
Breaking Down the Expression
1. Evaluate p → q:
- This translates to "If p is true, then q is true."
- Since p is true and q is false, p → q is false.
2. Evaluate ¬r:
- Since r is false, ¬r is true.
3. Evaluate ¬s:
- Since s is true, ¬s is false.
4. Evaluate ¬r ∨ ¬s:
- This is true (true ∨ false = true).
5. Combine Results:
- Now substituting back: ¬((p → q) ∧ (¬r ∨ ¬s)) becomes ¬(false ∧ true).
- This simplifies to ¬(false), which is true.
Conclusion
Since the expression is true for x = 11, it satisfies the condition. Hence, the integer x ≥ 2 that satisfies the original expression is:
11
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Community Answer
Let p, q, r, s represents the following propositions.p: x ∈ {8, 9...
Data:
p: x ∈ {8, 9, 10, 11, 12}
q: x is a composite number
r: x is a perfect square
s: x is a prime number
Explanation:
p ⇒ q means p̅ + q
As, q is composite number So, p ⇒ results in {8, 9, 10, 12}
As, r: x is a perfect square, ¬ r results in all the numbers which are not perfect square
So, ¬ r: {8, 10, 11, 12}
¬ s: {8, 9, 10, 12}
¬ r ∨ ¬ s = {8, 9, 10, 11, 12}
Now, (p ⇒ q) ∧ (¬ r ∨ ¬ s) = {8, 9, 10, 12}
¬ ((p ⇒ q) ∧ (¬ r ∨ ¬ s)) results in a number which is not present in ((p ⇒ q) ∧ (¬ r ∨ ¬ s))
¬ ((p ⇒ q) ∧ (¬ r ∨ ¬s)) will give {11}.
p: x ∈ {8, 9, 10, 11, 12}
q: x is a composite number
r: x is a perfect square
s: x is a prime number
Explanation:
p ⇒ q means p̅ + q
As, q is composite number So, p ⇒ results in {8, 9, 10, 12}
As, r: x is a perfect square, ¬ r results in all the numbers which are not perfect square
So, ¬ r: {8, 10, 11, 12}
¬ s: {8, 9, 10, 12}
¬ r ∨ ¬ s = {8, 9, 10, 11, 12}
Now, (p ⇒ q) ∧ (¬ r ∨ ¬ s) = {8, 9, 10, 12}
¬ ((p ⇒ q) ∧ (¬ r ∨ ¬ s)) results in a number which is not present in ((p ⇒ q) ∧ (¬ r ∨ ¬ s))
¬ ((p ⇒ q) ∧ (¬ r ∨ ¬s)) will give {11}.
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