If $A=\{f,l,o,w\}$ and $B=\{f,l,o,o,w\}$, then are A and B equal sets?

Question

If $A=\{f,l,o,w\}$ and $B=\{f,l,o,o,w\}$, are set $A$ and set $B$ equal?

I saw this example in my book. But what I had learnt is that two sets are said to be equal if they have same elements.

Here, set $A$ and $B$ have same elements but their cardinal numbers are not same. In this case are $A$ and $B$ called equal?

$A=B \leftrightarrow \forall x \ (x \in A \leftrightarrow x \in B)$. See Extensionality. — Mauro ALLEGRANZA, Mar 28 '17 at 11:53
Remember that ${\ldots}$ is a defined notation; if $A={f,l,o,w}$, then there's a formula in the language of first order logic with the binary connective $\in$, that says something about the terms $A,f,l,o,w$. What is it? And what does adding another $o$ between your brackets do to the corresponding formula? — Malice Vidrine, Mar 28 '17 at 11:57

hmakholm left over Monica · Answer 1 · 2017-03-28T14:39:41.653

5

A set does NOT remember how many times it contains each element. It can answer only "yes" or "no" to the question "is such-and-such one of your elements?" -- and the totality of those yes/no answers is everything the set is.

Two sets $A$ and $B$ are the same if any which thing is either in both $A$ and $B$ or in neither of them.

In your case, $f$, $l$, $o$, and $w$ are all both in $A$ and $B$, and everything that is not one of those four things are in neither of $A$ and $B$.

edited Mar 28 '17 at 14:39

answered Mar 28 '17 at 11:44

hmakholm left over Monica

291,611

means ? Are they equal or not? – pi-π Mar 28 '17 at 11:47
@LeonhardEuler: If you're who you claim to be, you should be able to figure that out yourself. I have given you a definition. Can you find anything that is not either in both $A$ and $B$ nor in none of them? – hmakholm left over Monica Mar 28 '17 at 11:48
Your first sentence makes it sound like there was a difference between putting an element into a set once or twice, but this is wrong. If I take it out once, it's gone, no matter how often I put it in. I know you tried to give a down-to-earth explanation, but this one doesn't really capture the truth. Maybe it would be better to say that a set doesn't care how often you put the set in. – martin.koeberl Mar 28 '17 at 12:13
@martin.koeberl: I think you must be missing the "not" in that first sentence. – hmakholm left over Monica Mar 28 '17 at 12:48
@HenningMakholm No, I'm not. While the sentence is correct, the wording "how many times it contains each element" sounds like there was a difference between "containing an element once" or containing an element twice", but just that the set doesn't know which one is true. This is wrong. – martin.koeberl Mar 28 '17 at 13:20
@martin.koeberl: What the sentence says is that the set does NOT remember any difference attributable to "how many times it contains each element". It would be wrong to claim that a set remembers such a thing, but that is the opposite of what the sentence states. – hmakholm left over Monica Mar 28 '17 at 13:42
@HenningMakholm I (still) know what the sentence tries to say and I still think it's misleading. A set contains a thing or it doesn't contain that thing. It cannot remember that it contains it twice because it cannot contain it twice. There is no higher truth the set doesn't have any access to where one sees that it contains that element actually two times. What the set knows about the element relation on it is all there is to know about the set. Your answer makes it sound like there was some other higher truth. – martin.koeberl Mar 28 '17 at 13:54
@martin.koeberl: You don't seem to contradict anything I write, except when you claim I'm writing something different from you. I'm not sure how I can help you from here. I've even edited the answer to capitalize the NOT you keep pretending isn't there. Do you want it bolded too? – hmakholm left over Monica Mar 28 '17 at 14:36
I just find the wording very unfortunate. There's no need to help me because I don't care whether your answer is good or not (just to the extent that it might mislead people). No need to get unfriendly. – martin.koeberl Mar 28 '17 at 14:40
@martin.koeberl: "No need to get unfriendly" -- when you have spent a lot of comments now telling me haughtily that SOMETHING THE ANSWER EXPLICITLY DENIES is false? – hmakholm left over Monica Mar 28 '17 at 14:44
Let us continue this discussion in chat. – martin.koeberl Mar 28 '17 at 14:47

Arnaldo · Accepted Answer · 2017-03-28T12:05:16.417

2

We can say, by definition that $$A=B \Leftrightarrow A\subset B \text{ and } B \subset A$$ what is true in your case.

It doesn't matter how many times you put the same element in the set.

For example, if $A=\{a\}$ and $B=\{a,a\}$ we can see that if we take any element of $A$, which is $a$, this element we can find in $B$ and then $A\subset B$.

Similarily, if we take any element of $B$, which is always $a$, we also can find it in $A$ and then $B\subset A$.

So $A=B$.

edited Mar 28 '17 at 12:05

answered Mar 28 '17 at 11:45

Arnaldo

21,758

4

I fear that does not really clear it up -- because the OP is probably also uncertain whether ${o,o}\subseteq{o}$. – hmakholm left over Monica Mar 28 '17 at 11:46
@HenningMakholm: I think the OP got it. – Arnaldo Mar 28 '17 at 12:06

score 1 · Answer 3 · answered Mar 28 '17 at 11:54

Two sets, $X$ and $Y$, are equal if every element of $X$ is an element of $Y$ and each element of $Y$ is an element of $X$.

In your case, every element of $A$ is also an element of $B$, and every element of $B$ is also an element of $A$. Therefore, the two sets are equal.

score 1 · Answer 4 · answered Mar 28 '17 at 11:56

1

Two sets are equal if they have same elements. If in any set any element is repeating. Still both are equal.

They are not equal if any set containing element that other set doesn't have.

answered Mar 28 '17 at 11:56

Kanwaljit Singh

8,820

score 0 · Answer 5 · answered Mar 28 '17 at 12:07

0

They are the same: each of $f$, $l$, $o$, and $w$ is in each of $A$ and $B$ but there are no other elements of $A$ and $B$. Repetitions are ignored.

answered Mar 28 '17 at 12:07

Shaun

47,747

If $A=\{f,l,o,w\}$ and $B=\{f,l,o,o,w\}$, then are A and B equal sets?

5 Answers5