Meaning of the word 'Relate' in context of Course Objectives
The word 'Relate', when provided as a link in Course Objectives, was referred to in the Startup and Orientation. In this document, as promised, we discuss the word more extensively.
A more precise definition in terms of standard set notation is given a little later, but we begin with a discussion in terms of some examples that should be familiar to anyone with general secondary-level knowledge of mathematics.
The basic idea of the word 'Relate' is the following:
If we know a bunch of things, maybe we can put some of them together and figure out new things.
If we know a bunch of things, maybe we can pick just a few of them that we could have used to figure out all the others (if we didn't already know them).
... and maybe, if we do use a few things to figure out the others, we'll find that we were wrong about some of them.
Once we know how to Relate a bunch of things there are a couple of things we can do:
We can use our understanding to solve problems.
We can often 'grow' our bunch of things to include new things related to the old things.
We can invent new things to relate to bunch.
The basic idea is, so far, pretty abstract. So let's look at a few specific examples
First example:
Here's a small bunch of things:
{one number, another number, the sum of the two numbers}
Just to be really clear we might add that the 'sum' of two numbers is what we get when we add them. (This statement is actually sort of backward because the idea of 'sum' comes before the idea of addition, but we aren't going to worry about that right now)
There are only three things in this bunch.
If we know two of the things, then we can figure out the third. Specifically:
if we know the first number and the second, we can add them to get the third
if we know the first number and the sum of the two numbers, we can subtract the first number from the sum and get the second
if we know the second number and the sum of the two numbers, we can subtract the second number from the sum and get the first
We also understand that if we only know one of the three things, we don't have enough information to find the other two.
So we know everything that can be known about how these three things are related.
This is what it means to 'Relate' a bunch of things.
Notice that in my first example I wrote the 'bunch of things' as the list {one number, another number, the sum of the two numbers}, separated by commas and enclosed between a set of braces { }. You will recall that in mathematics when we write our 'bunch of things' in this way, we can call it a 'set'. The 'things' we listed are called elements of the set.
In the above example, then, we have shown that we can 'Relate' the set {one number, another number, the sum of the two numbers}, meaning that
each individual thing can be found other things in the bunch, or 'set'
and
that it's possible to list all the 'relationships', meaning all the ways individual things can be combined to get new ones.
So here's a first attempt to define the word 'Relate':
We can 'Relate' a bunch of things if:
we know every way two or more things can be combined to get other things
we know every combination that can be used to figure out the whole bunch
we know that every single thing in the bunch can be found from other things in the bunch
If we combine two or more things to get another thing, we say that our combination 'generates' that other thing.
Saying the same thing in terms of sets, we say that we can 'Relate' a set if:
we know what elements of the set can be generated from any given subset of the set
we know every subset capable of generating the entire set
we know that every element of the set can be generated by some subset not containing that element
We could get a little more formal with the definition, but we're not trying to construct a formal mathematical theory here. All we're trying to do is get a handle on a useful way to think about what is means to know stuff.
Second example:
Here's another small bunch of things:
{one number, another number, the sum of the two numbers, the product of the two numbers}
There are four things in this bunch. (We can also say that there are four elements in this set.)
Can some of these things be related to others?
Obviously if we know the two numbers, we can add them to find their sum and multiply them to find their product.
So from the two numbers, we can find everything else in the bunch.
We say that the set
{one number, another number}
generates the set
{one number, another number, the sum of the two numbers, the product of the two numbers}
Also, if we know the first number and the sum, we can subtract to find the second number. Then knowing the two numbers we can multiply them to find their product.
So from one number and the sum of the two numbers, we can again find the whole bunch.
We say that the set
{one number, sum of the two numbers}
generates the set
{one number, another number, the sum of the two numbers, the product of the two numbers}
How many other ways can you find to combine two or more of these things to find the rest?
It turns out that we can prove that any two of these things can be used to find the other two.
But to do so we have to prove that if we know the product and the sum of two numbers, we can find the two numbers.
You can probably figure out what two numbers have a sum of 6 and a product of 8.
But can you figure out what two numbers have a sum of 23 and a product of 107?
That requires some algebra, and not everyone who has had algebra can do this.
An even fewer can prove that it's always possible, given the sum and product of two numbers, to find the numbers.
So what have we learned about the idea of 'Relate'?
It's easy to see that someone who knows only arithmetic can understand a lot about how to 'Relate' the set
{one number, another number, the sum of the two numbers, the product of the two numbers}.
However you have to know some stuff that goes beyond arithmetic, and be fairly clever as well (or be within a few days of a test on the topic), to prove that any two of these things are enough to figure out the the other two.
So
Some bunches of things can be partially related, some can be completely related, but not everyone can completely understand all the relationships.
That's the way it is with most knowledge.
In this course the better you understand the relationships, the better you are likely to do.
To help out
We list most of the sets to be related.
You master the relationships by practicing them, mostly by solving problems.
More technical discussion (in the language of set theory):
The definition implied here is actually a little more restrictive than its use in practice. It would be possible to repair this, but the resulting definition would be more involved than necessary.
In the language of set theory we can say the following:
The set {one number, another number, sum of the two numbers}, defined for any two real numbers, has the characteristic that if we know the values of any two of the three numbers in the set, we can find the third. Any literate fifth-grader understands how to relate two numbers and its product (including the fact that the product can be divided by one of the numbers to get the other), and therefore understands why it is so (though not many fifth-graders have the sophistication to present this in the form of a complete logical argument).
The set {one number, another number, sum of the two numbers, product of the two numbers}, again defined for any two real numbers, also has the characteristic that if we know the values of any two of the three numbers in the set, we can find the third. While a literate fifth-grader perfectly understands most of the relationships, and typical high school graduate should have the knowledge to explain this in full, only a small minority of college graduates could actually do so.
A 'relationship' among two or more things is a way of using some of those things to figure out another.
If two or more things can be used to figure out another, then they are said to 'generate' the latter.
To 'Relate' a set is to know for any given element of the set all the ways it could be generated from subsets, and given a subset to know element it generates.
For some sets this is achievable.
For many sets it is possible to relate all but a few of its subsets and/or elements.
for any given subset of S we know which elements of S can be generated by the elements of that subset
we know every subset capable of generating the entire set
and, perhaps too restrictively: we know that every element of the set can be generated by some subset not containing that element
A subset S of U is said to generate an element x of U if
We can 'Relate' a set U if
A bulleted list of items, subject to the word 'Relate', is understood to constitute a set U. Specifically each item in the list is taken to be an element of the set U, and U contains no element not in the list.
The power set P(U) contains 2^(n(U)) elements; i.e., there are 2^n subsets of a set which contains n elements.
A 10-element set therefore contains over 1000 subsets, and a 20-element set over a million.
It is theoretically possible but neither practical nor useful to list, for every element of U, the subset of the power set which generates it.
It is, however, possible and sometimes useful to list at least one 'minimal' subset which generates each element of U.
It is theoretically possible to define an assessment metric in terms of the level of relational knowledge demonstrated by the student when solving problems.
A reasonable metric could not generally be a linear function of, say, the proportion of the number of elements of a given set the student appears (through sampling) to be capable of generating. There might exist sets for which a linear metric happens to be reasonable, but any effort to impose a condition of linearity would put the assessment before the subject.
It would be possible over time to correlate the ability to generate various items or subsets with less well-defined grading rubrics, and the associated database might at least provide useful information and reflection on various aspects of assessment.