Equivalence relation for $X \sim Y \iff X \cap T = Y \cap T$

Question

For the question

Let $T$ be a fixed subset of a nonempty set $S$. Define the relation $\sim$ on power set of S by $$X \sim Y \iff X \cap T = Y \cap T$$ Show that $\sim$ is an equivalent relation. In particular let $S = \{1,2,3,4,5\} \text{ and } T = \{1,3\}$

Is $\{1,2,4\} \sim \{1,4,5\}$ and equivalent relationship. What about $\{1,2,4\} \sim \{1,3,4\}$?

The book just gives the answer as true for the first and false for the second. I want to know why my answers are wrong because I get true for both and how to get the right answer.

$\{1, 2, 4\} \sim \{1,4,5\}$ is an equivalence relation because:

• Reflexive: $X \cap T = X \cap T$ is true
• Symmetric: $X \cap T = Y \cap T \implies Y \cap T = X \cap T$ is true
• Transitive: $\{1, 2, 4\} \cap \{1,3\} = \{1,4,5\} \cap \{1,3\} \land \{1,4,5\} \cap \{1,3\} = Z \cap \{1,3\} \implies \{1,2,4\} = Z$ is true

$\{1, 2, 4\} \sim \{1,3,4\}$ is an equivalence relation because:

1. Reflexive is true (see above)
2. Symmetric: $\{1, 2, 4\} \cap \{1,3\} = \{1,3,4\}\cap\{1,3\}$ is false since $\{1\} \neq \{1,3\}$ since the premise is false, the implication is true
3. Transitive: $\{1, 2, 4\} \cap \{1,3\} = \{1,3,4\} \cap \{1,3\} $ is false since $\{1\} \neq \{1,3\}$. Since the premise is false, the implication is true

Answer 1

Honestly, I've been as confused as you are right now when I first saw this stuff. So let's go over it step by step.

First, we need to show the relation $\sim$ satisfies the demands of an equivalence relation. Your reflexivity and symmetricity checks are correct. For transitivity you have, for some reason, replaced $X,Y,T$ with some fixed subsets. To fix this, assume $X\cap T = Y\cap T$ and $Y\cap T = Z\cap T$. Is is then true that $X\cap T = Z\cap T$?

Once we've established that $\sim$ is an equivalence relation, we fix sets $S,T$ and we are asked whether $$\{1,2,4\} \sim \{1,4,5\}\quad\mbox{or}\quad \{1,2,4\}\sim \{1,3,4\}. $$ To check this, we simply check the condition we defined the relation $\sim$ with. Do the following hold? $$\{1,2,4\}\cap \{1,3\} \overset{?}= \{1,4,5\}\cap \{1,3\} \quad (\text{yes}) \\ \{1,2,4\}\cap \{1,3\} \overset{?}= \{1,3,4\}\cap \{1,3\}\quad (\text{no})$$

Can you pick it up from here?

Answer 2

You're mixing up two different things:

• Whether the relation $\sim$ is an equivalence relation or not.

• Whether two particular elements are related by $\sim$ or not.

It does not make sense to ask whether "$\{1,2,4\}\sim\{1,4,5\}$" is an equivalence relation or not, becasue "$\{1,2,4\}\sim\{1,4,5\}$" is not a relation. It is just a claim -- and this happens to be true because the $\sim$ relation you have defined does indeed say that the element $\{1,2,4\}$ is related to the element $\{1,4,5\}$. But it makes no sense to ask, for example, whether the fact that $\{1,2,4\}\sim\{1,4,5\}$ "is reflexive" or not.

I think you're being confused by the fact that $\{1,2,4\}$ and $\{1,2,5\}$ are sets of numbers. However, as far as the relation $\sim$ is concerned those sets are elements, and its properties as a relation do not depend on the incidental fact that the elements it speaks about happen to be sets.

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For the exercise I would strongly suggest doing the second part first, to get a feeling for what $\sim$ does in two particular cases.

Then consider whether the entire $\sim$ is an equivalence relation or not.

Answer 3

No $\{1,2,4\} \not\sim \{1,4,5\}$ since $\{1,2,4 \} \cap \{1,3 \} = \{1 \} \neq \{1,3\} = \{1,3,4 \} \cap \{1,3 \}.$

Answer 4

Something got garbled in your statement of the question. You write “Is $\{1,2,4\} \sim \{1,4,5\}$ an equivalent relationship.” First off, what does “equivalent relationship” mean? Your work makes it seem like you think it means “equivalence relation,” but that doesn’t make sense, because $\{1,2,4\} \sim \{1,4,5\}$ is a single true-or-false statement. Asking if a single true-or-false statement is an equivalence relation is like asking whether the number $5$ is a triangle. It’s not even the right kind of thing. I think the question is whether $\{1,2,4\} \sim \{1,4,5\}$ and $\{1,2,4\} \sim \{1,3,4\}$ are true or false. Your attempt doesn’t work, because the properties of relexivity, symmetry, and transitivity make no sense to ask about for a single statement like $\{1,2,4\} \sim \{1,4,5\}$.