1. Logic and Proof help!

I'd like to know if I did these problems correctly:

1.Give an example of a partition of N into 3 subsets.
ANSWER: Let s={A,B,C) where A={x ∈ N:2x},B={x ∈ N:4x-1} and C={1}

2.For a real # r, define S<subscript>r</subscript>to be the interval [r-1,r+2]. Let A={1,3,4}. Determine ∪∝∈A S∝ and ∩∝∈ A S∝
ANSWER: ∪∝∈A S∝=(-∞,∞) and ∩∝∈ A S∝=0(empty set)

I'm not sure how to answer this next problem, I'm sure if I see part "a" done I can do the other parts that I didn't post!

3.For a real number r, define A<subscript>r</subscript>={r^2} B<subscript>r</subscript> as the closed interval[r-1,r+1] and C<subscript>r</subscript> as the interval (r,∞) For S={1,2,4}

a) ∪∝∈S A∝ and ∩∝∈ S A∝ btw this is part of the problem not the answer >_>

2. Welcome to the forum.

1.Give an example of a partition of N into 3 subsets.
ANSWER: Let s={A,B,C) where A={x ∈ N:2x},B={x ∈ N:4x-1} and C={1}
This is a wrong set-builder notation. For example, in A={x ∈ N:2x}, what is to the right of : must be a property that is either true or false for each x ∈ N, whereas you have a number there. If you mean A = {2x : x ∈ N}, B = {4x - 1 : x ∈ N} and C = {1}, then this is not a partition because 5 is not in A ∪ B ∪ C.

2.For a real # r, define S<subscript>r</subscript>to be the interval [r-1,r+2]. Let A={1,3,4}. Determine ∪∝∈A S∝ and ∩∝∈ A S∝
ANSWER: ∪∝∈A S∝=(-∞,∞) and ∩∝∈ A S∝=0(empty set)
It is true that $\displaystyle\bigcup_{r\in\mathbb{Z}}S_r=(-\infty,\infty)$ ( $\mathbb{Z}$ is the set of integers); however, you are only asked to find $\displaystyle\bigcup_{r\in A}S_r=S_1\cup S_3\cup S_4=[0,3]\cup[2,5]\cup[3,6]$. The intersection $S_1\cap S_3\cap S_4$ is not empty either.

3.For a real number r, define A<subscript>r</subscript>={r^2}... For S={1,2,4}

a) ∪∝∈S A∝ and ∩∝∈ S A∝
We have $\displaystyle\bigcup_{r\in S}A_r=A_1\cup A_2\cup A_4=\{1\}\cup\{4\}\cup\{16\}=\{1,4,16\}$. Also, $\displaystyle\bigcap_{r\in S}A_r=\emptyset$.

A couple of remarks concerning notation. In plain text, it is customary to express subscripts and superscripts using _ and ^. If you want to typeset nice formulas, you can put text inside the $$...$$ tag. You can double-click on formulas in this post to see their source code, or you can click "Reply with code" to see the code for the whole post.

The symbol ∝ is usually used to denote some binary relations; it is strange to use it as a letter (like x or r).