Question

Let \(A=\{1,2,3,\ldots ,10\}\). Consider the following statements:
\(p:\forall x\in A,x^{2}-x>0\)
\(q:\exists x\in A\), such that \(x+10=20\)
Then the truth values of \(p\) and \(q\) are respectively ____

• A. \(T,T\)
• B. \(T,F\)
• C. \(F,T\)
• D. \(F,F\)

Hint:

Check whether the condition holds for every element in the first statement and for at least one element in the second.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
A statement with \(\forall\) is true only if it holds for every element.
A statement with \(\exists\) is true if at least one element satisfies it.

Step 2: Check p
For \(x=1\), \(x^{2}-x=0\), which is not greater than \(0\).
So \(p\) is false.

Step 3: Check q
We need \(x+10=20\), so \(x=10\). Since \(10\in A\), \(q\) is true.

Step 4: Conclusion
The truth values of \(p\) and \(q\) are F and T.

Final Answer:
p is false and q is true. \[ \boxed{\text{(C) }F,T} \]