Test 1 #5 (Version A)

Use the definition of big- $Theta$ notation to determine whether $f(x)=x^2$ is $\Theta(x^3)$ .

To show that $f(x)=x^2$ is $\Theta(x^3)$ , we would have to show that $f(x)=x^2$ is $O(x^3)$ and that $f(x)=x^2$ is $\Omega(x^3)$ .

However, we can see that $f(x)=x^2$ is not $\Omega(x^3)$ , as follows:

(This will be a proof by contradiction). Assume that $f(x)=x^2$ is $\Omega(x^3)$ . This means that there exists a pair of positive constants $(C,k)$ such that, for all $x>k$ , we have that $|x^2| \geq C|x^3|$ .
Since $k>0$ and $x>k$ we have that $x^2>0$ .
This means that we are assuming that $x^2 \geq Cx^3$ for all $x>k$ .
It also means that we can safely divide both sides of the inequality by $x^2$ without changing it.
This gives that $1 \geq Cx$ for all $x>k$ .
Since $C>0$ , this means that $\frac{1}{C} \geq x$ for all $x>k$ .
Whatever $C$ is, it’s just a constant, so $\frac{1}{C}$ is also just a constant. So this is saying that for all large values of $x$ , $x$ stays below the constant $\frac{1}{C}$ . This is a contradiction, so our initial hypothesis, that $f(x)=x^2$ is $\Omega(x^3)$ , must have been false.

Since $f(x)=x^2$ is not $\Omega(x^3)$ , $f(x)=x^2$ cannot be $\Theta(x^3)$ .

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One Response to Test 1 #5 (Version A)

Kate Poirier says:

March 27, 2016 at 5:18 pm

Many of you had the correct answer, but didn’t justify it. To show that $f(x)=x^2$ is *not* $\Omega(x^3)$ , you have to show that there is *no possible pair* $(C,k)$ such that the condition in the definition is satisfied. It’s not enough to pick a value for $C$ and show that it has no partner $k$ . It’s also not enough to pick a value for $C$ and a value for $k$ and show that they do not satisfy the condition in the definition.

A few of you included a proof that $f(x)=x^2$ is $O(x^3)$ , which is true but not necessary since $f(x)=x^2$ is not $\Omega(x^3)$ .

One Response to Test 1 #5 (Version A)

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