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

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

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

b. $\mathbb{Z}$ is the identity of $S(\mathbb{Z})$ with respect to $\cap$ so the only subset $A$ of $\mathbb{Z}$ with an inverse with respect to $\cap$ is $\mathbb{Z}$, 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 $S(\mathbb{Z})$ be the set of all subsets of $\mathbb{Z}$.
a. Which subsets $A$ of $\mathbb{Z}$ have inverses for $\cup$? What are they?

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

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

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