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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 Quote:

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 $\displaystyle \displaystyle\bigcup_{r\in\mathbb{Z}}S_r=(-\infty,\infty)$ ($\displaystyle \mathbb{Z}$ is the set of integers); however, you are only asked to find $\displaystyle \displaystyle\bigcup_{r\in A}S_r=S_1\cup S_3\cup S_4=[0,3]\cup[2,5]\cup[3,6]$. The intersection $\displaystyle S_1\cap S_3\cap S_4$ is not empty either. Quote:

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 $\displaystyle \displaystyle\bigcup_{r\in S}A_r=A_1\cup A_2\cup A_4=\{1\}\cup\{4\}\cup\{16\}=\{1,4,16\}$. Also, $\displaystyle \displaystyle\bigcap_{r\in S}A_r=\emptyset$.