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

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

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?

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.

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.

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

• Does this mean it's not reflexive?
– Sam
Commented Apr 6, 2019 at 17:55
• The relation $\sim$ is an equivalence relation. Hence it is obviously reflexive. Here the point is that whether the sets $\{1,2,4 \}$ and $\{1,3,4 \}$ are related by the equivalence relation $\sim$ or not. To observe that you must go through the definition of that relation $\sim.$ It says $A$ is related to $B$ by the relation $\sim$ iff $A \cap T = B \cap T.$ So in order to check whether $\{1,2,4 \}$ is related to $\{1,3,4 \}$ via the relation $\sim$ you need to verify whether $\{1,2,4 \} \cap \{1,3 \} = \{1,3,4 \} \cap \{1,3 \}$ or not. Commented Apr 6, 2019 at 18:02

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 $$\{1,2,4\} \sim \{1,4,5\}$$.

• is the question then just asking if the two sets belong in the same equivalence class?
– Sam
Commented Apr 6, 2019 at 18:09
• Correct you are! Commented Apr 6, 2019 at 18:09