# Thread: Inverses With Respect To Union and Intersection

1. ## Inverses With Respect To Union and Intersection

I'm not sure about my solution to the following problem and I'd appreciate it if someone could check it.

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Let $\displaystyle S(\mathbb{Z})$ be the set of all subsets of $\displaystyle \mathbb{Z}$.
a. Which subsets $\displaystyle A$ of $\displaystyle \mathbb{Z}$ have inverses for $\displaystyle \cup$? What are they?

b. Which subsets $\displaystyle A$ of $\displaystyle \mathbb{Z}$ have inverses for $\displaystyle \cap$? What are they?
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Here is my solution:

a. $\displaystyle \{\}$ is the identity of $\displaystyle S(\mathbb{Z})$ with respect to $\displaystyle \cup$ so the only subset $\displaystyle A$ of $\displaystyle \mathbb{Z}$ with an inverse with respect to $\displaystyle \cup$ is $\displaystyle \{\}$, namely itself.

b. $\displaystyle \mathbb{Z}$ is the identity of $\displaystyle S(\mathbb{Z})$ with respect to $\displaystyle \cap$ so the only subset $\displaystyle A$ of $\displaystyle \mathbb{Z}$ with an inverse with respect to $\displaystyle \cap$ is $\displaystyle \mathbb{Z}$, namely itself.

2. Originally Posted by rualin
I'm not sure about my solution to the following problem and I'd appreciate it if someone could check it.

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Let $\displaystyle S(\mathbb{Z})$ be the set of all subsets of $\displaystyle \mathbb{Z}$.
a. Which subsets $\displaystyle A$ of $\displaystyle \mathbb{Z}$ have inverses for $\displaystyle \cup$? What are they?

b. Which subsets $\displaystyle A$ of $\displaystyle \mathbb{Z}$ have inverses for $\displaystyle \cap$? What are they?
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Here is my solution:

a. $\displaystyle \{\}$ is the identity of $\displaystyle S(\mathbb{Z})$ with respect to $\displaystyle \cup$ so the only subset $\displaystyle A$ of $\displaystyle \mathbb{Z}$ with an inverse with respect to $\displaystyle \cup$ is $\displaystyle \{\}$, namely itself.

b. $\displaystyle \mathbb{Z}$ is the identity of $\displaystyle S(\mathbb{Z})$ with respect to $\displaystyle \cap$ so the only subset $\displaystyle A$ of $\displaystyle \mathbb{Z}$ with an inverse with respect to $\displaystyle \cap$ is $\displaystyle \mathbb{Z}$, namely itself.
Good Job.