If a = b, and b = c, then ...?

[KlipKlop]

Hi

I’ve been searching online and I can find this (transitive property of equality):

If a = b and b = c, then a = c

but not this:

If a = c and b = c, then a = b

So, my question is:

Is the second statement valid? To my mind, it seems that Euclid’s first common notion (things equal to the same thing are equal to each other) should apply to both statements.

Thank you

 

[Deleted member 4993]

Hi

I’ve been searching online and I can find this (transitive property of equality):

If a = b and b = c, then a = c

but not this:

If a = c and b = c, then a = b

So, my question is:

Is the second statement valid? To my mind, it seems that Euclid’s first common notion (things equal to the same thing are equal to each other) should apply to both statements.

Thank you

If a = b and b = c, then a = c

We know if x = y then y = x (reflexive property)

Then rewriting the given statement:

If b = a and b = c, then a = c

 

[KlipKlop]

Thanks. Is it ok if I ask a similar question?

If a = b and a = c, what is the relationship of b to c?

 

[Mrspi]

Thanks. Is it ok if I ask a similar question?

If a = b and a = c, what is the relationship of b to c?

This looks mighty similar to another you just discussed.....

You've got two things, b and c, and each of those things is equal to a If two things are each equal to the same thing, what must be true of those two things?

Or, we could apply the symmetric property of equality, and rewrite a = b as b = a. Now, we have this: b = a and a = c, and you can use the transitive property.....

Either way, we come up with the same relationship between b and c.

 

[KlipKlop]

Here's why I asked the second question ... What if a, b, and c are not numbers, for example:

dogs (a) are (=) animals (b)
and dogs (a) are (=) pets (c)
then animals (b) are (=) pets (c)

which isn't necessarily true, because not all animals can be domesticated.

Ok, that's not a very good example but it might illustrate my problem with: if a=b and a=c, then b=c. By not using numbers, but rather objects or concepts, am I talking about a different "realm" than mathematics?

 

[lookagain]

Here's why I asked the second question ... What if a, b, and c are not numbers, for example:

dogs (a) are (=) animals (b)
and dogs (a) are (=) pets (c)
then animals (b) are (=) pets (c)

which isn't necessarily true, because not all animals can be domesticated.

Ok, that's not a very good example but it might illustrate my problem with: if a=b and a=c, then b=c.
By not using numbers, but rather objects or concepts, am I talking about a different "realm" than mathematics?

But dogs don't equal animals. Dogs are a subset of animals, so your set-up is wrong to begin with.

 

[Ishuda]

Here's why I asked the second question ... What if a, b, and c are not numbers, for example:

dogs (a) are (=) animals (b)
and dogs (a) are (=) pets (c)
then animals (b) are (=) pets (c)

which isn't necessarily true, because not all animals can be domesticated.

Ok, that's not a very good example but it might illustrate my problem with: if a=b and a=c, then b=c. By not using numbers, but rather objects or concepts, am I talking about a different "realm" than mathematics?

If you are going to extend to 'non-numbers' then you are going to have to be more careful about what you mean by equal. By dogs=animals do you mean that all animals are dogs as well as all dogs are animals which is just the transitive property of the equal sign and numbers. If not, then you need to define what you mean by equal in this case. The same applies to the dogs=pets.

Typically, in situations like this, by dogs are animals one means that dogs are a subset of the group of animals,
{dogs} \(\displaystyle \subset\) {animals}
Similarly dogs are pets
{dogs} \(\displaystyle \subset\) {pets}
These two statements do not imply any particular relationship between pets and animals. A pet may or may not be an animal [remember the pet rock craze some time ago or, at a later date, the Furby craze] and, as you mentioned, an animal may or may not be a pet.

 

[Steven G]

Hi

I’ve been searching online and I can find this (transitive property of equality):

If a = b and b = c, then a = c

but not this:

If a = c and b = c, then a = b

So, my question is:

Is the second statement valid? To my mind, it seems that Euclid’s first common notion (things equal to the same thing are equal to each other) should apply to both statements.

Thank you

I am not sure why you are confused with the 2nd statement. 1st note that b=c and c=b are the same statement. So your 2nd statement is the same as If a=c and c=b, then a=b. If we replace c with B and b with C, then we get If a=B and B=C, then a=C. But this is the 1st statement.
Here is the proof of the 1st statement.
a=b means a-b=0 and b=c means b-c=0. Adding a-b=0 and b-c=0 yields a-c =0 or a=c.