Intro 3. Verbs and Adverbs

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3. Verbs and Adverbs
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. We will hereafter adopt the term verb instead of (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 (like a primitive verb) have both monadic and dyadic cases; 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.

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