the definition of the word "equation" in math

Question

Google give the following definition of the word "equation":

"In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true."

Is really an "equation" is a statement? I think that it more close to a "predicate". I will be happy if someone can give an exact (but still simple) definition of the term "equation"

P.s. In all cases, I mean equation with unknown(s), not equal. Thanks!

Answer 1

I don't like Google's definition. It's much too narrow for serious mathematics.

Dealing formally with statements of the form $$ P = Q $$ is subtle. I won't go into it here. (But do read Barry Mazur's When is one thing equal to some other thing, as @Shaun suggests.)

Informally, an equation (in mathematics) is a statement that asserts that two mathematical objects are the same. So $$ 1 + 1 = 2 $$ is an equation. The left member is another way to write the number $2$, not an instruction to add $1$ and $1$.

The equation $$ 1 + x = 2 $$ says those two expressions describe the same number. It's implicitly a suggestion that you should "solve for $x$" - find the numerical values (if any) that make the assertion true. That's Google's sense.

Sometimes equations are appear in definitions: we can specify $f$ as the function given by the equation $$ f(x) = 1 + x $$ for real numbers $x$.

Answer 2

Algebraically equality of two objects is often defined if they are "the same". We define this in different contexts, but mostly we define it together with the concept of an order relation. Some examples:

In set theory: $$ A \subseteq B \land B \subseteq A \implies A=B$$

Since a set is defined by the elements it contains, if one set is contained in the other and vice versa, clearly they must contain the same elements. Here we see equality because a set is uniquely determined by its elements.

In ordered fields, using the trichotomy axiom ($\mathbb{Q}$ is an ordered field, not completely ordered though, $\mathbb{R}$ is): $$ a\leq b \land b\leq a \implies a=b$$

In group theory: let $e$ be the identity element and $a$ an arbitray element in the group: $$ e \cdot a = a \cdot e=a$$ This statements tells us that applying the identity elements does not do anything, it also does not matter from which side we apply it. All these examples only further support the idea we see equality as two things being "the same". We can even define the relation "=" for any two numbers, clearly:

$$ a=a$$ $$ a=b \implies b=a$$ $$ a=b \land b=c \implies a=c $$ So equality is an equivalence relation.

Another fun example there are equally as much even-sized subsets as odd-sized of a given set:

$$ 0=(1-1)^n = \sum_{k=0} ^n binom{n}{k} (-1)^n $$ Here the binomial tells us how many $k-sized$ subsets there are of a set $n$, we notice that all possible subsets are either even or odd by a partition of the natural numbers into these two sets. We can show there exists a bijection between the two so they have the same cardinality. We can split the sum up over odd and over even $k$ and notice that these two terms cancel, there must be the same amount of terms and thus there must be the same amount of subsets for all $n$. So there are equally as much off-subsets as even -subsets.

Surprisingly when two objects are "the same" in algebra, but have a different representation, we speak of the idea of an "isomorphism", which is also an interesting concept. This means we can identify both the structure and the elements uniquely. All very interesting ideas, definitely a nice question to think about, thank you!