Okay so the question is: Rewrite each of the following statements in the two forms "∀x, if _____then____ "and “∀ ___x, ____ "(without an if-then). a.The square of any even integer is even. b.Every computer science student needs to take data structures.
After looking online and trying for hours on how to do this myself, I looked up the answer so I would have a baseline to work towards. But I found another problem that is similar to the one I'm working on and the answer to it. Here it is:
Consider the statement "The square of any odd integer is odd." Rewrite the statement in the form n, (Do not use the words "if" or "then.") Rewrite the statement in the form n, if then (Make sure you use the variable n when you fill ill each of the second two blanks. $$ ∀z ∈ ℤ, \text{ if }∃n ∈ ℤ st.z= 2n+1 \text{ then }∃m ∈ ℤ \text{ st }z^2= 2m + 1 $$ $$ ∀z ∈ ℤ\text{ st }∃n ∈ ℤ z=2n+1, ∃m ∈ ℤ \text{ st }z^2= 2m + 1 $$
I would just like to know how to really break down this answer and apply it to what I've learned also where did that m come from? Will I need something similar in my problem?
This is assuming you want help interpreting those statements. If you answer my comment and indicate otherwise, I will modify this answer to better fit the question!
The first statement follows:
For all integers z, if there exists some element n within the integers so that z may be written as 2n+1 then there will exist some element m within the integers so that $z^2=2m+1$.
Incase you are unaware, the definition of an odd number is: a number $x$ is odd if there exists an integer $n$ so that $x=2n+1$
So,considering this definition, you can see that the first statement here is saying something along the lines of:
If you square an odd integer the result will be a new odd integer. Which is true. It follows from the fact that an odd number multiplied by an odd number is always odd.
The second statement
Basically says the same thing as the first. The difference is that it is closer to:
For all odd integers...
How this applies to your problem
You want to write equivalent statements for even numbers. Just be aware of the definition of even. Compare that to the definition of odd. The changes you should make to those statements once you know the difference should make it trivial to modify.