Equivalence relation, pleas ehelp

[Frankenstein143]

Please help because I still do not understand the reflexive relation here.
Here the Problem:

X= world population

x~y: <=> x and y have a Birthday on 23. April.

Is this statement equivalent?

I tried to solve it and came to the results:

A1: it is reflexive because: if person x has a Birthday on 23. April then he has on 23. April Birthday (I mean it is the same person)

A2. It is symmetric because: if person x and y have on 23. April Birthday then have y and x Birthday on 23 April (it is just in different order)

A3: It is transitive because If x and y have a Birthday on 23. April and y and z have Birthdays on 23. April then x and y have also a Birthday on 23. April

BUT! I was said it is false, and the reason is: it is not the case that

for all

And I still do not understand why is it false. For me, my results are right. I have difficulties understanding Abstract things and therefore I make it a little bit more realistic:
Assume that:
our world population is X:={Anna, Berta, Caesar, Dennis, Elena}
According to the Definition of Reflexive: every Element of X relates to itself. So what does it mean?
Let us take Anna since she is an element of X. So if Anna has a Birthday on 23. April then she has Birthday on 23. April because Anna relates to herself.

For me it is reflexive.

Please explain somebody to me as simple as possible

 

[pka]

Please help because I still do not understand the reflexive relation here.
Here the Problem: X= world population
x~y: <=> x and y have a Birthday on 23. April.
Is this statement equivalent?
BUT! I was said it is false, and the reason is: it is not the case that

for all

For a relation to be reflexive it must apply to each element in the upon which it is defined.
I for one has my birthday on Jan. 26 therefor \(pka{\bf\large\cancel\sim} pka.\)
Thus, I am in the world but the relation does not apply to me.
So the relation is not reflexive. It is symmetric & transudative.

 

[Frankenstein143]

But it sounds so strange? I mean, the definition says: every Element of X relates to itself.
That means if you have a birthday on Jan. 26 then you relate to yourself. I have a birthday on 2. Mai, then I relate to myself.
Isn't that what the definition tells us that every x of X relates to itself?

 

[pka]

Here the Problem: X= world population, x~y: <=> x and y have a Birthday on 23. April.

But it sounds so strange? I mean, the definition says: every Element of X relates to itself.
That means if you have a birthday on Jan. 26 then you relate to yourself. I have a birthday on 2. Mai, then I relate to myself.
Isn't that what the definition tells us that every x of X relates to itself?

Read the statement of the question.
It says that world's population is the domain of discourse and \(\Large {\bf x\sim y} \iff x~\&~y\) have a birthday on the twenty-third day of April.
That means that two people are related by that relation if and only if their birthday is on 23. April. XXXX. otherwise the people are not related.
It is impossible that everyone has 23. April as their birthday.
If the relation were \(x\sim y\iff~x~\&~y \text{ have the same day and month as a birthday }\) then that is a reflexive relation for all the reasons you state.

 

[lex]

[MATH]x\text{~}x \Longleftrightarrow x[/MATH] has a birthday on 23 April and [MATH]x[/MATH] has a birthday on 23 April i.e. [MATH]\Longleftrightarrow x[/MATH] has a birthday on 23 April
So assuming that not everyone in the world was born on 23 April then there is an [MATH]x \in X: x \not \text{~} x[/MATH]

 

[Steven G]

You have X = the world population. Now x~y if and only if x and y were born on April 23.
Now take someone random from X, the word population. This person, let's call them z, is related to themselves, ie z~z, if and only if z has their birthday on April 23. This may be true for z, but it is not certainly true for everyone in X.