Find a counterexample to the following statements among the different propositions.
" f\left(x\right)=x^2-3x+2 has no real positive root."
A counterexample is an example that contradicts the proposed statement.
The proposed statement is " f\left(x\right)=x^2-3x+2 has no real positive root".
Therefore a positive real root is required to contradict the statement.
f \left(2\right) = 2^ 2 + -3 \cdot 2 +2=4-6+2=0
2 is a positive real root of f\left(x\right) hence it is a counterexample.
"2 is a root of f " is a counterexample to this statement.
"All odd numbers are prime."
A counterexample is an example that contradicts the proposed statement.
The proposed statement is "all odd numbers are prime".
Therefore a composite odd number is required to contradict the statement.
9 is a composite odd number, hence it is a counterexample.
9 is composite and an odd number hence it is a counterexample.
"All even numbers can be written as a power of two, such as 2=2^1 and 4=2^2."
A counterexample is an example that contradicts the proposed statement.
The proposed statement is "all even numbers can be written as a power of 2".
Therefore an even number which has an a odd factor in it's prime factorization is required to contradict the statement.
6 is an even number, and 6=2 \cdot 3, so it can not be written as a power of 2. Hence 6 is a counterexample.
6 is even and can not be written as a power of 2.
"All prime numbers are odd."
A counterexample is an example that contradicts the proposed statement.
The proposed statement is "all primes are odd".
Therefore an even number which is prime is required to contradict the statement.
2 is an even number, and it's prime. Hence 2 is a counterexample.
2 is even and prime, hence it is a counterexample.
" f\left(x\right)=x^2-3x+2 has no real positive root."
A counterexample is an example that contradicts the proposed statement.
The proposed statement is " f\left(x\right)=x^2-3x+2 has no real positive root".
Therefore a positive real root is required to contradict the statement.
f \left(6\right) = 6 ^ 2 + -2 \cdot 6 -24 = 36-12-24 = 0
6 is a positive real root of f\left(x\right) hence it is a counterexample.
6 is a real positive root of f(x).
"All animals living in the ocean are fish."
A counterexample is an example that contradicts the proposed statement.
The proposed statement is "All animals living in the ocean are fish".
Therefore a non-fish animal living in the ocean is required to contradict the statement.
Whales live in the ocean and they are not fish. Hence whales are a counterexample.
Whales live in the ocean and they are not fish is a counterexample.
" f \left(x\right) = x ^ 2 -x-12 has no real negative roots."
A counterexample is an example that contradicts the proposed statement.
The proposed statement is " f \left(x\right) = x ^ 2 -x-12 has no real positive root".
Therefore a positive real root is required to contradict the statement.
f \left(-3\right) = \left(-3\right) ^ 2 -\left(-3\right) -12 = 9+3-12=0
-3 is a negative real root of f\left(x\right) hence it is a counterexample.
-3 is a negative real root of f(x).