Worksheet 1: Above and Beside

GeomLab uses the constants man and woman to represent stick-figure pictures of a man and a woman:

man picture
woman picture

Whenever an expression and a picture are shown like this, we mean that GeomLab will show the picture as the value of the expression. You should try to work out why the expression creates the picture that is shown, and (if you like) check that it does so by typing the expression into GeomLab.

Two operators that can be used to combine these images are $ and &. As illustrated in the two examples below, $ means "beside", and & means "above":

man $ woman picture
man & man picture

What would woman $ man look like? Sketch the picture in the space below:

woman $ man picture

Try typing woman $ man into GeomLab and see if you were right.

Geomlab has a number of other constant pictures, including tree, so typing the expression (man $ woman) $ tree produces the following image:

(man $ woman) $ tree picture

What happens when you type man $ (woman $ tree) into GeomLab? Sketch here the image that is produced:

man $ (woman $ tree) picture

You should find that both pictures are the same!

In the first example above, (man $ woman) is worked out (or evaluated) first, producing a picture of a man standing beside a woman. Then the $ tree part of the expression puts a tree to the right of this picture.

Similarly, in the second example (woman $ tree) is evaluated first, producing a picture of a woman and a tree standing next to each other. Then the man $ part of the expression is evaluated, placing a man to the left of this picture.

We can see that the same picture was produced in the two examples above, so the two expressions we used are the same (or equivalent). Since it's true for all pictures p, q and r that

(p $ q) $ r = p $ (q $ r)

we say that $ is associative. Since p $ q is not always the same as q $ p, we say that $ is not commutative.

You can try for yourself in GeomLab similar examples to those above, but replacing $ with &. You should see that & is also associative but not commutative.

Try typing man $ (woman & tree) into GeomLab. You should get an image that looks like this:

man $ (woman & tree) picture

Something new is happening here. When GeomLab combines two pictures using $, their sizes are adjusted so that the pictures have the same height, without changing the shape of either picture.

In man $ woman, they have the same height already, so no adjustment is needed. However, in man $ (woman & tree), the picture woman & tree is rather tall and thin, so it must be reduced in size so that its height is equal to the height of the other man.

You will see that the picture produced above is different from the one given by the expression (man $ woman) & tree:

(man $ woman) & tree picture

What happens if you put no brackets in? Try typing man $ woman & tree into GeomLab and sketch the image that results:

man $ woman & tree picture

You should see that the expression man $ woman & tree produces the same image as the expression (man $ woman) & tree. This happens because a $ operation in an expression is evaluated before an & operation, just as multiplication is evaluated before addition in an ordinary, mathematical expression. For example, if you were asked to work out the answer to 3 × 4 + 5, you would first work out 3 × 4 and then add 5 to the result, giving an answer of 17. We say that × binds more tightly than +, and similarly that $ binds more tightly than &. When GeomLab works out what picture to draw for man $ woman & tree, it evaluates man $ woman first, and evaluates the & tree part afterwards.

Remember that although $ binds more tightly than &, brackets will still be evaluated first:

man $ ((man $ tree) & woman) picture

Can you work out what expression gives this picture?

picture

Now look at this picture:

(man $ tree) & (woman $ man)

(man & woman) $ (tree & man)
picture

Both expressions give the same picture (try it!), so the expressions are equivalent.

If the pictures p, q, r and s all have the same shape (as they do here), then the two expressions

(p $ q) & (r $ s)

and

(p & r) $ (q & s)

produce the same picture.

But if the component pictures have different shapes, this is not always true. For example, applying the rule to ((man & woman) $ man) & (woman $ tree) we get the following pictures:

((man & woman) $ man) & (woman $ tree) picture
((man & woman) & woman) $ (man & tree) picture

We can see that the two pictures are not the same, since some of the figures have been changed in size (or scaled) differently. If you are good at algebra, you might like to try to work out exactly when the equation

(p $ q) & (r $ s) = (p & r) $ (q & s)

is true for four pictures p, q, r and s. A hint: it has to do with the ratios between the widths and the heights of the pictures. The scaling rule of GeomLab means that these ratios do not change when the pictures are scaled up or down; both sides of the equation will give the same result if the boundaries between the pictures meet in a point.

In this sheet, we have seen that it makes to work with formulas whose values are not numbers but pictures. We can have constants that stand for fixed pictures, and we can have operators that combine pictures to make new ones. Just as in ordinary algebra, it makes sense to think about the relative priority of the operators: just as convention dictates that multiplication is done before addition where they appear together in a formula, we can make the convention that $ is done before &.

In ordinary algebra, some equations are true no matter what values we put for the variables they contain: for example, the equation a + (b + c) = (a + b) + c expresses the fact that addition is associative. Similarly, we can look for equations that always hold between formulas in our new kind of algebra.

Looking beyond our little world of pictures, there are many places in programming and computer science where the algebraic idea of having a set of values with operations on them is relevant. For example, tables of information in databases can be combined with algebraic operations that match the values in a column of one table with values that appear in a different column of another table. This is a good way of organising a database system so that people who use it in their programs are insulated from the way the data is stored.

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