Is this Proof of Equality Correct?

[Mathematicsresear]

Homework Statement

Prove the following:

If x=y and y=z then x=z.

Now, this seems very obvious, and it is without a doubt correct. However, I am curious as to if the following proof is correct.

Homework Equations

The Attempt at a Solution

Assume x does not equal to z, so that means two cases:

Case 1) x>z, so z-x<0 and since x=y, z-y<0 which is a contradiction.

Case 2) z>x, so x-z<0 and since y=z or x=y ( I can use either), that means x-y<0 which is a contradiction. Therefore it has to be that if x=y and y=z then x=z.

 

[fresh_42]

Homework Statement

Prove the following:

If x=y and y=z then x=z.

Now, this seems very obvious, and it is without a doubt correct. However, I am curious as to if the following proof is correct.

Homework Equations

The Attempt at a Solution

Assume x does not equal to z, so that means two cases:

Case 1) x>z, so z-x<0 and since x=y, z-y<0 which is a contradiction.

Case 2) z>x, so x-z<0 and since y=z or x=y ( I can use either), that means x-y<0 which is a contradiction. Therefore it has to be that if x=y and y=z then x=z.

I'm not sure whether you can use the ordering. If $x,y,z$ weren't numbers, the transitivity of $"="$ would still be given. Transitivity is the name of the assertion you want to prove. It somehow transports the burden simply to somewhere else. Normally transitivity is part of a definition, a requirement. Thus the question is: what does $"="$ mean? Do you have a definition for it?

You have in principle shown: $x=y \wedge y=z \wedge x\neq z\,\Rightarrow\, y\neq z$, because we can substitute $y$ for $x$ and then have a contradiction. But isn't this substitution already the transitivity which we want to show?

If you have the ordering as a given, plus the uniqueness of $0$, then your proof is o.k. because you translated the problem to the uniqueness of $0$.

 

[Mathematicsresear]

I'm not sure whether you can use the ordering. If $x,y,z$ weren't numbers, the transitivity of $"="$ would still be given. Transitivity is the name of the assertion you want to prove. It somehow transports the burden simply to somewhere else. Normally transitivity is part of a definition, a requirement. Thus the question is: what does $"="$ mean? Do you have a definition for it?

You have in principle shown: $x=y \wedge y=z \wedge x\neq z\,\Rightarrow\, y\neq z$, because we can substitute $y$ for $x$ and then have a contradiction. But isn't this substitution already the transitivity which we want to show?

If you have the ordering as a given, plus the uniqueness of $0$, then your proof is o.k. because you translated the problem to the uniqueness of $0$.

= is the binary relation on the set of natural numbers, for instance, 1=1, 2=2 and n=n.

 

[fresh_42]

= is the binary relation on the set of natural numbers, for instance, 1=1, 2=2 and n=n.

Yes, but as a binary relation it should have some properties which defines it. Usually it is reflexivity $x=x$, symmetry $x=y \Rightarrow y=x$ and transitivity $x=y \wedge y=z \Rightarrow x=z$. But as you should show the latter, there must be something else we can use instead. One possibility is your way with the calculation rules, but then we already use the relation. So it's a bit confusing.

E.g. your proof substitutes $z$ by $y$. But if this is allowed, then transitivity follows directly by this substitution and we don't need the difference. That's why I'm asking what is given.