Equal sign meaning in equations vs functions

Question

My kids are learning about functions for the first time, and I was hoping for a little clarity concerning what the equal sign means in a variety of contexts.

I have checked out, and understood parts of, the available discussion on the differences between equations and functions. (As a philosopher it was much easier to follow the "word explanations" than the explanations in "math notation." My intro to logic class was at least 100+ years ago, and didn't help much.)

I would like to offer my understanding of the different ways the equal sign seems to be used (with examples), and would love constructive feedback.

In equations, the equal sign seems to be used to denote the following things; (these descriptions are my own words, I'm sure there are "real" names for what I am trying to describe, also I don't really understand the difference between "equals" and "is equivalent to"):

1. tautologies: $5 = 5$;
2. equivalent "values": $4 = 8/2$ (the value four is equivalent to the value of eight halves);
3. equivalent expressions: $2x/4 = x/2$;
4. value/ expression of a given variable: "if $x = 2$...";
5. value of an unknown variable, after you know what the "answer" is; the value that makes the equation true "Solve for $x$; [math stuff]; $x = 14$."

In functions, the equals sign seems to have a different use. It seems to me that in "function notation" the equal sign is used more to mean "therefore" or the "then" part of an "if then" conditional. But even here there seems to be several uses.

1. $f(x) = 7x + 3$, seems to say "for function $f$, if the input is $x$, then the output will be $7x + 3$." So the equal sign relates input to output.
2. But $f(x) = 7x + 3$ also seems to relate input to the rule ("relation"?) that governs the output values. So in this case it means: "given this input, do this, to calculate the output."
3. (given function $f$ in #1) "$f(2) = 17$" relates a specific input value (2) to a specific output value (17).
4. Also, if given a function $f$ (from #1), $f(x) = 24$, seems to represent the question "what input (into function f) leads to an output of 24?" Or in other words: output + rule, leads to input.

The one overlap I have managed to find between equations and functions, is that the equal sign seems to be able to represent a "conditional" for both equations and functions.

• $f(x) = 7x + 3$ seems to say: "for $f$, if $x$, then $7x + 3$."

• $f(2) = 17$ seems to say: "for $f$, if 2, then 17."

• $f(x) = 17$ seems to say: "for $f$, if (something unknown), then 17". Or: "for $f$, what input leads to output of 17?"

• $8/4 = 2$ means: "if you have eight fourths, then you (also) have two wholes."

• $x = 3$, as a "given" means: "given equation 'something', if you use 3 as the value of x, then...".

• $x = 5$, as a solution, means: "given equation 'something', if you use 5 for the value of $x$, then the equation will be true."

So what do you think? I do really think the equal sign and its variety of uses, is the source of my confusion. Does any of what I have said make sense? Please help! (And thanks!)

Answer 1

In the comments, people have pointed out various explanations of the different kinds of "equality" that can be encountered in mathematics. But I think most (if not all) of these responses are deeper than you need to look in order to resolve your difficulties.

In my view, there is only one meaning of the symbol $=$ in all of the examples given in the question. The meaning of $=$ is that the expression on the left of the $=$, when fully evaluated to its fundamental mathematical object (which is a number in every one of the examples given) is precisely identical to the mathematical object that is obtained by fully evaluating the expression on the right of the $=$.

In short, when you evaluate what you see on each side of the $=$, you get the exact same thing.

I think the difficulty here is not the meaning of $=$, which is always the same, but the interpretation of symbols such as $x$, which has to be inferred informally from context in some mathematical texts because we are too lazy to make it explicit.

For example, you might come across a passage like this:

The function $f$ defined by $f(x) = 7x + 3$ ...

This is a kind of mathematical cliché that in the context of the study of real numbers actually means to say the following:

The function $f$ that maps real numbers to real numbers in such a way that for any real number, if you let $x$ represent that real number and use $x$ as the "input" of $f$, then the "output" of $f$ (that is, the number represented in this case by the expression $f(x)$) is the number that you get when you evaluate $7x + 3.$

That is, each of these statements (the first one by convention, the second one more explicitly) has stated the facts $f(2) = 17,$ $f(3)=24,$ $f(4)=31,$ and an infinite number of other such facts all in one swell foop.

I think it's obvious why the shorter way of saying this is more popular than the long way.

On the other hand, again in the context of the study of real numbers, we could write something like this:

Let $f(x) = 1 - x^2.$ Solve for $x$ in $f(x) = 7x + 3.$

The difference between "$f(x) = 7x + 3$" here and "$f(x) = 7x + 3$" in the previous statements is not the meaning of $f(x) = 7x + 3$ but rather is what the surrounding text says about when the statement $f(x) = 7x + 3$ is true. When we write, "The function $f$ defined by $f(x) = 7x + 3$," we are saying that the statement $f(x) = 7x + 3$ is true for all real numbers, but when we write, "Solve for $x$ in $f(x) = 7x + 3$," we are asking when the statement $f(x) = 7x + 3$ is true. Specifically, we are asking which values of $x$ (if any) will make the statement $f(x) = 7x + 3$ true in the context of a previously-stated definition of the function $f.$

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Perhaps your question is not so much about the meaning of $=$ as it is about the uses of equations. There are certainly a lot of ways mathematics texts use equations and (if you're not familiar with the mathematical style of writing) it's not always clear how you're supposed to understand the equations.

This will be a lengthy response because you brought up many good examples and I will try to examine each one carefully. Also, I am trying to anticipate uses of equations that you may not yet have encountered but which would likely cause confusion later.

It's also possible that I am belaboring the point in parts of this response because the only difference between your understanding and mine is that you expressed yours differently. Please bear with me in those cases.

Consider these specific uses of $=$ from the question:

1. $5 = 5$

The expression consisting of the single symbol $5$ on the left hand side evaluates to a number. (In many contexts we would say it is a number, but when we're trying to carefully parse the meaning of the symbol $=$, it's worth noting that in this context the symbol $5$ may be better viewed as a name for a number rather than the number itself.)

The symbol $5$ on the right hand side also evaluates to a number. And it's the same number (surprise, surprise). So the equality statement is true.

2. $4 = 8/2$

The symbol $4$ evaluates to a number. The expression $8/2$ evaluates to a number. These two numbers are the same, so the statement is true.

3. $2x/4 = x/2$

If $x$ were a number and we knew what number $x$ was, we would be able to evaluate $2x/4$ and obtain a number as the result. Likewise with $x/2.$ But to understand what we're supposed to do with this equation, we need to look at its context.

Usually the context of an equation like this is in the statement of an identity: we are asserting that the equation is true for every number $x$ within some set of numbers, such as all the real numbers. I think this is what you meant by "equivalent expressions". But you can also encounter problems like the following:

Let $2x/4 = x/2.$ Solve for $x.$

(I used the word "like" loosely here; usually the problem will start with a more complicated equation than this, or even multiple equations.) If you are studying real numbers, the answer to this problem is that every real number can be used as a value of $x$ to make the equation true. A solution set is the set of all numbers we can use as the evaluation of $x$ in order to make a statement (such as $2x/4 = x/2$) true; for this problem, if we are studying real numbers, the solution set is the set of all real numbers.

4. if $x = 2$ ...

This says, "If the symbol $x$ evaluates to the same thing that $2$ does ... ." It could be a way of "trying on" a particular evaluation of $x$ in some other expression. In the context, "If $x = 2$ then $f(x) = 17,$" it's a roundabout way of saying $f(2)=17.$

5. "Solve for $x$; [math stuff]; $x = 14$."

I think you have the right idea about this. The big clue is the phrase "solve for $x$".

This example says that there is only one possible way to evaluate $x$ in order to make [math stuff] true, and that $x$ must evaluate to the same thing as $14.$

It is not inevitable that the conclusion of a "solve for $x$" exercise is an equation of the form "$x = \text{something}$". The conclusion might be "no solution" or it might be something like $x \in \{4, 14\}.$ (You can write the latter as "$x=4$ or $x = 14$" if you want to avoid using the notation of sets.)

Now let's consider the examples using functions:

1. $f(x) = 7x + 3$ [...] relates input to output.
2. But $f(x) = 7x + 3$ [...] "given this input, do this, to calculate the output."

In the context of a function definition (see the first part of this answer), both interpretations are correct. The equation tells us how the "input" of the function relates to its "output". In particular, by putting an expression for the output of the function on one side of the $=$ symbol and an expression we can evaluate algorithmically on the other side, it gives us a procedure for calculating all outputs of the function.

There are other ways to define functions. For example, to define $f$ as the principal square root function, if we haven't already defined that function, we could write

For $x \geq 0,$ $(f(x))^2 = x$ and $f(x) \geq 0.$

3. (given function $f$ in #1) "$f(2) = 17$" relates a specific input value (2) to a specific output value (17).

This is a good interpretation. You can leave out the part about the "given" function and it is still a true interpretation; it merely doesn't say what the rest of the definition of function $f$ is.

4. Also, if given a function $f$ (from #1), $f(x) = 24$, seems to represent the question "what input (into function f) leads to an output of 24?" Or in other words: output + rule, leads to input.

Also a good interpretation. There should usually be another clue about what you're supposed to do about the value of $x$, such as "find $x$" or "solve for $x$."

• $f(x) = 7x + 3$ seems to say: "for $f$, if $x$, then $7x + 3$."
• $f(2) = 17$ seems to say: "for $f$, if 2, then 17."
• $f(x) = 17$ seems to say: "for $f$, if (something unknown), then 17". Or: "for $f$, what input leads to output of 17?"

I would explain these a little differently, because a number is not a statement that can be true or false; "if $2$ then $17$" is mathematically meaningless.

This answer would become ridiculously long if it got into the question, "What is a function in mathematics, really?" But it may help to know that the expression $f(x)$ means "the output value of $f$ that corresponds to the input value $x$." In this light, you could say (informally) that $f(2)=17$ means that if the input of $f$ is $2$ then the output of $f$ is $17$.

Note that depending on context, $f(x) = 17$ could be a problem to solve for $x,$ or it could be the definition of the function $f$ whose output is always $17$ for every input.

• $8/4 = 2$ means: "if you have eight fourths, then you (also) have two wholes."

If we're talking about apples, eight fourths is not the same as two wholes; the whole apples may still be good to eat tomorrow, but the fourths will go brown quickly. But if we're just talking about numbers, yes, you have the same number as the value on both sides.

• $x = 3$, as a "given" means: "given equation 'something', if you use 3 as the value of x, then...".
• $x = 5$, as a solution, means: "given equation 'something', if you use 5 for the value of $x$, then the equation will be true."

I couldn't have said it better.

In summary, it's not so much about "what does this equation mean" but rather, "OK, here's a statement of an equation, but how am I expected to use it?" The clues to answer that question are unfortunately subtle, so subtle that sometimes people post specific arithmetic questions on this site that other people are unable to answer due to inability to see those clues. Sometimes you just have to read a textbook for a while to figure out what clues that author tends to give (including when the absence of a "clue" phrase is itself a clue).

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I should note that as a student of philosophical logic, you would have been introduced to contexts in which the form of an expression is as important as, if not more important than, a number to which you might evaluate the expression (if you even can evaluate it). What I'm trying to say in this answer is that your children's textbooks almost surely do not treat the expressions in equations that way. I would expect to see them use the same single definition of an equation that I described above, and that all the differences in the significance of the equations is due to what the book does with the equation in relation to the other things found in the same passage or exercise.

If your "kids" are currently doing graduate-level work then I could be wrong about all this.

Answer 2

Long question, short answer (since already the comments link to details).
The meaning of $=$ is apparent from the context. Some programming languages don't like to guess and force the use of different signs, $:=$ and $=$, or even $==$.

BTW, the $=$ is kind of an obsolescent model me think, because even dated pocket calculators added $=0$ at the end of an entered equation. Thus A-1/A-1 suffices completely to compute the Golden Ratio.