Proving Equality in Math: Solving the Question of Two Equaling a Third

[zeromodz]

If two are equal to a third, do they all equal each other? Can you prove this?

 

[nicksauce]

Depending on what you mean by "two" and "a third", this fact follows from the transitive property.

 

[Mentallic]

I'm going to assume it means if two numbers are equal to 1/3, are the numbers equal?

Well, not necessarily. For them to be equal that means each number has to be 1/6, but of course you can give each number any other value which still satisfies that they add to 1/3.

 

[Mark44]

The question seems to be ambiguous. I interpret it as nicksauce does, to mean "if two numbers are equal to a third number, then all three are equal."

 

[Redbelly98]

I'm going to assume it means if two numbers are equal to 1/3, are the numbers equal?

You're interpreting this to mean the sum of two numbers has a value equal to 1/3. Mark44 and nicksauce interpret it to mean that two numbers are each, themself, equal to a third number.

 

[Mentallic]

Oh yeah that seems like a much more reasonable interpretation

 

[Redbelly98]

It's better than my initial thought of "2 = 1/3"

 

[disregardthat]

It's called the principle of explosion. If you grant a false statement, then you can logically prove that any statement is true. Hence your system "explodes" so that the truth values of all statements are 1 (and 0). So if 2 = 1/3, then a = b for any numbers a and b. This can however be proved by mere algebraic manipulation. From 2 = 1/3 you will get 6 = 1, so 5 = 0, or 1 = 0, and from there a - b = 0, so a = b.

EDIT: Oh, after reading the other comments, I realize that the question really was whether a = c, b = c implies a = b. That has been explained.

 

[slider142]

Your original sentence fragment is missing its subject. I assume you mean "If two [objects] are equal to a third [object], ..."
This is true as long as your definition of "equal" shares the transitive property, which is included in the usual mathematical definition of an equivalence between objects.
Specifically, the transitive porperty states that if the relationship between three objects, {a, b, c}, is denoted with the sign "=" such that a = c and b = c, then it follows that a = b.
However, I do not know what your definition of equality is, or what 3 objects you are referring to.
It is quite possible that you are talking about a different kind of equality, which is mathematically more like placing objects into sets.
That is, "A chicken is an animal.", and "A dog is an animal." do not enjoy the transitive property that "A chicken is a dog."