# Intro03

In the sentence %a of Section 2, the % “acts upon” a
to produce a result, and %a is therefore analogous
to the notion in English of a verb acting upon a noun or pronoun.
(or in addition to) the mathematical term function used thus far.

The sentence +/ 1 2 3 4 is equivalent
to 1+2+3+4; the adverb / applies to
its verb argument + to produce a new verb
whose argument is 1 2 3 4, and which is
defined by inserting the verb + between the items of its argument.
Other arguments of the insert adverb are treated similarly:

```   */b=:2 7 1 8 2 8
1792
<./b
1
>./b
8
```

The verb resulting from the application of an adverb may
due to its two uses, the adverb / is called either
insert or table. In the present instance of /
the dyadic case produces a table. For example:

```   2 3 5 +/ 0 1 2 3
2 3 4 5
3 4 5 6
5 6 7 8
```

The verbs over=:({.;}.)@":@, and by=:' '&;@,.@[,.]
can be entered as utilities (for use rather than for immediate study),
and can clarify the interpretation of function tables such as
the addition table produced above. For example:

```   a=: 2 3 5
b=: 0 1 2 3
a by b over a +/ b
+-+-------+
| |0 1 2 3|
+-+-------+
|2|2 3 4 5|
|3|3 4 5 6|
|5|5 6 7 8|
+-+-------+
b by b over b </ b
+-+-------+
| |0 1 2 3|
+-+-------+
|0|0 1 1 1|
|1|0 0 1 1|
|2|0 0 0 1|
|3|0 0 0 0|
+-+-------+
```

Exercises

 3.1 Enter d=: i.5 and the sentences st=: d-/d and pt=: d^/d to produce function tables for subtraction and power. 3.2 Make tables for further functions from previous sections, including the relations < and = and > and lesser-of and greater-of. 3.3 Apply the verbs |. and |: to various tables, and try to state what they do. 3.4 The transpose function |: changes the subtraction table, but appears to have no effect on the multiplication table. State the property of those functions whose tables remain unchanged when transposed. Answer: They are commutative 3.5 Enter d by d over d!/d and state the definition of the dyad ! . Answer: ! is the binomial coefficient or outof function; 3!5 is the number of ways that three things can be chosen from five.