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# equality

In any set $S$, the equality, denoted by “$=$”, is a binary relation which is reflexive, symmetric, transitive and antisymmetric, i.e. it is an antisymmetric equivalence relation on $S$, or which is the same thing, the equality is a symmetric partial order on $S$.

In fact, for any set $S$, the smallest equivalence relation on $S$ is the equality (by smallest we mean that it is contained in every equivalence relation on $S$). This offers a definition of “equality”. From this, it is clear that there is only one equality relation on $S$. Its equivalence classes are all singletons $\{x\}$ where $x\in S$.

The concept of equality is essential in almost all branches of mathematics. A few examples will suffice:

$\displaystyle 1+1$ | $\displaystyle=$ | $\displaystyle 2$ | ||

$\displaystyle e^{{i\pi}}$ | $\displaystyle=$ | $\displaystyle-1$ | ||

$\displaystyle\mathbb{R}[i]$ | $\displaystyle=$ | $\displaystyle\mathbb{C}$ |

Remark 1. Although the four characterising properties, reflexivity, symmetry, transitivity and antisymmetry, determine the equality on $S$ uniquely, they cannot be thought to form the definition of the equality, since the concept of antisymmetry already contains the equality.

Remark 2. An equality (equation) in a set $S$ may be true regardless to the values of the variables involved in the equality; then one speaks of an identity or identic equation in this set. E.g. $(x+y)^{2}=x^{2}+y^{2}$ is an identity in a field with characteristic $2$.

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## Comments

## Equalities, set theory, and branch without equalities

This is an important entry that has been missing for so long. I have to rate it high even though I have a couple of small concerns with it:

1. Is there a branch of mathematics where the concept of equality or inequality is completely irrelevant? If not, "almost all" should be changed to "all."

2. Doesn't the set of birds include only animals with "feathers, a beak with no teeth, the laying of hard-shelled eggs, a high metabolic rate, a four-chambered heart, and a lightweight but strong skeleton" (according to Wikipedia)? So bats would be excluded from this set because they don't lay eggs. If I'm right about this, it would be more correct to say that "the intersection of the set of mammals and the set of flight-capable animals is the set of birds." If this is too awkward, surely a better example can be found to illustrate equalities in set theory.

## Re: Equalities, set theory, and branch without equalities

Good day.

> 1. Is there a branch of mathematics where the concept of

> equality or inequality is completely irrelevant? If not,

> "almost all" should be changed to "all."

Unfortunately, I know none, so I cannot be of help.

> 2. Doesn't the set of birds include only animals with

> "feathers, a beak with no teeth, the laying of hard-shelled

> eggs, a high metabolic rate, a four-chambered heart, and a

> lightweight but strong skeleton" (according to Wikipedia)?

> So bats would be excluded from this set because they don't

> lay eggs. If I'm right about this, it would be more correct

> to say that "the intersection of the set of mammals and the

> set of flight-capable animals is the set of birds." If this

> is too awkward, surely a better example can be found to

> illustrate equalities in set theory.

Birds are not mammals, so the phrase should be "the intersection of the set of non-mammals and the set of flight-capable animals is the set of birds."

But this includes insects, so perhaps "the intersection of the set of non-mammals, the set of vertebrates, and the set of flight-capable animals" should go...

Best regards,

Silvio

## Re: Equalities in set theory

Correct me if I'm wrong, but I think they're looking for a set that intersected with the set of mammals gives the set of bats. A bat is a mammal but is not a bird. And not all birds can fly (e.g., penguins, ostriches, etc.)

If they can work out these technicalities, then it will be an example that makes perfect sense mathematically. Or they could just define an universe in which a bird is any vertebrate flying animal (this is almost like the mathematician who fenced himself in and defined inside as outside).

Or maybe just use sets of numbers, e.g., "The intersection of the primes and the intersection of the even numbers equals a set with only one element, 2."

## Re: Equalities in set theory

Good day.

> Correct me if I'm wrong, but I think they're looking for a

> set that intersected with the set of mammals gives the set

> of bats. A bat is a mammal but is not a bird. And not all

> birds can fly (e.g., penguins, ostriches, etc.)

Well, I had interpreted the message as saying that they wanted to define the set of birds, not the set of bats.

Moreover, no mammal is a bird and no bird is a mammal, so saying "a bat is a mammal but it is not a bird" gives no more information than either "a bat is a mammal" or "a bat is not a bird".

On the other hand, the observation that not all birds can fly is noteworthy---I don't remember exactly the definition I gave and am too lazy to check it now, but it can probably be adjysted by replacing "flying" by "winged". (All birds have wings, even vestigial.)

> Or maybe just use sets of numbers, e.g., "The intersection

> of the primes and the intersection of the even numbers

> equals a set with only one element, 2."

I've started thinkin' that could be the simplest thing to do...

Best regards,

Silvio

## Re: Equalities in set theory

>> Or maybe just use sets of numbers, e.g., "The intersection

>> of the primes and the intersection of the even numbers

>> equals a set with only one element, 2."

>I've started thinkin' that could be the simplest thing to do...

The example of the real numbers being a subset of the complex numbers would be even simpler notationally, so I've gone ahead and added it. I do regret losing the bats and birds examples, but perhaps it is really way more trouble than it's worth.