Dfns

A dfn (pronounced “dee-fun” with a very short “u” sound) is a way of writing functions in APL. It starts and ends with curly braces {}, has a right argument with ⍵ (omega) and an optional left argument ⍺.

      3{⍺}5      ⍝ ⍺ is the (optional) left argument
3     
      {⍵}'apl'   ⍝ ⍵ is the right argument
apl      
      {⍺}5       ⍝ Calling a dyadic function monadically results in an error
VALUE ERROR
      {⍺}5
      ∧            
      3{⍵}       ⍝ Calling a function without a right argument results in an error
SYNTAX ERROR: Missing right argument
      3{⍵}
       ∧

Assignment

Names are assigned with the left arrow ←. We say “name gets [function or array]”.

      one←1      
      three←3      
      equals←=      
      plus←+      
      four←4      
      four equals one plus three   ⍝ 1 means true, 0 means false
1

We can use a name in the same line in which it is defined. In production code it is best to avoid this unless an expression is very short.

      ⍝ Squared numbers divided by the sum of squares 
      
      squared÷+/squared←¯1 0 1 2*2 
0.1666666667 0 0.1666666667 0.6666666667`

Problem Set 2: Dfns and Assignment

Utility Functions
1. Without using the tally ≢ or shape ⍴ functions, create a function named Tally which returns the number of elements in a vector.
```
       Tally 1 2 4 523 1 2 454
 7
```
2. Create a function named Mean which returns the mean average of a numeric vector.
```
       Mean 1 2 4 523 1 2 454
 141
```
3. Using ⌊ (“floor” i.e. round down), create a function IsDivisibleBy which returns 1 if ⍺ is divisible by ⍵ and 0 otherwise.
```
       15 IsDivisibleBy 5
 1        
       15 IsDivisibleBy 6
 0       
       12 IsDivisibleBy ⍳12
 1 1 1 1 0 1 0 0 0 0 0 1
```

What Remains

Mod is a dyadic function which returns the remainder after its right argument ⍵ number is divided by the left argument ⍺.

       16÷5  
 3.2              
       5 Mod 16   ⍝ 3 5s and 1 left over
 1              
       0.2×5      ⍝ 0.2 5s left over
 1              
       12 Mod 30
 6        
       13 Mod 56
 4        
       5 Mod 30   ⍝ 5 goes into 25 exactly 6 times      
 0

Write the Mod function as a dfn.

Write a function named Split which takes a fractional number and returns two numbers: its integer and fractional parts.
```
       Split 0
 0 0        
       Split ¯10
 ¯10 0        
       Split 7.62
 7 0.62
```

What Was In That Vector Again?

You should have a variable named ⎕AVU in your workspace, from Part 1.
1. How many even numbers are there in ⎕AVU?
  
  Answer: 132
2. What percentage of numbers in ⎕AVU are odd numbers?
  
  Answer: 48.4375
3. What percentage of numbers in ⎕AVU are strictly negative?
  
  Answer: 0
4. What percentage of numbers in ⎕AVU are strictly positive?
  
  Answer: 99.609375
5. What do you notice about the percentage of strictly positive and negative numbers?
Prime Time

A prime number is divisible only by itself and 1.

Write a dfn which returns 1 if its argument is prime and 0 otherwise.
```
       IsPrime 21
 0
       IsPrime 17
 1
```