MATH220 April 2005
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Question 01

The Intermediate Value Theorem states:
 For any real numbers a < b, and for any function
 Failed to parse (syntax error): {\displaystyle f: [a,b] \to \mathbb{R},}
 if f is continuous on the interval [a,b ], then for every real number y between f(a) and f(b), there exists a real number c in the interval [a,b ] such that f(c)=y.
Write down the negation of this statement. (Don't worry that the negation will be a false statement  just negate the statement correctly).

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1

A theorem is always a statement of the type A $\Rightarrow$ B and since we have the equivalence
 $\displaystyle A\Rightarrow B\equiv \lnot A\lor B$
we can deduce that the negation of such a statement is
 $\displaystyle {\begin{aligned}\lnot (A\Rightarrow B)&\equiv \lnot (\lnot A\lor B)\\&\equiv A\land \lnot B\end{aligned}}$
Or in plain English: the negation of A implies B is when A is true and B isn't.

Hint 2

The previous hints claims that a theorem is of the type A $\Rightarrow$ B. Can you identify each of A and B?

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
 If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
 If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

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The Intermediate Value Theorem states:
 For any real numbers a < b, and for any function
 Failed to parse (syntax error): {\displaystyle f: [a,b] \to \mathbb{R},}
 if f is continuous on the interval [a,b ], then for every real number y between f(a) and f(b), there exists a real number c in the interval [a,b ] such that f(c)=y.
We can usually spot implications in theorems by the use of the word then. So here we have that the statement A is
 $\displaystyle A$ = ƒ is a continuous function on the interval [a,b ], with a < b.
and the conclusion of the theorem is statement B
 $\displaystyle B$ = for all real numbers y between ƒ(a) and ƒ(b) there exists a real number c in the interval [a, b ] such that ƒ(c) = y.
And so the Intermediate Value Theorem is simply the statement
 $\displaystyle A\Rightarrow B$
And its negation is the statement
 $\displaystyle A\land \lnot B$
We'll need the negation of statement B for this:
 $\displaystyle \lnot B$ = there exists a real number y between ƒ(a) and ƒ(b) such that for all real number c in the interval [a, b ] we have ƒ(c) ≠ y.
And so the negation is
 There exists a continuous function ƒ on the interval [a,b ], with a < b such that there exists a real number y between ƒ(a) and ƒ(b) such that for all real number c in the interval [a, b ] we have ƒ(c) ≠ y.

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