]box on -style=max -trains=tree -fns=on
┌→─────────────────────────────────────┐ │Was ON -style=max -trains=tree -fns=on│ └──────────────────────────────────────┘
= (Equal sign)¶= (Equal to)¶= is equal and returns a boolean (true/false). 1 means true, 0 means false.
1 = 1
1
= works elementwise and will broadcast as needed.
1 = 1 2
┌→──┐ │1 0│ └~──┘
1 = 1 1
┌→──┐ │1 1│ └~──┘
We can also compare characters
'Banana' = 'aaaaaa'
┌→──────────┐ │0 1 0 1 0 1│ └~──────────┘
'Banana' = 'a'
┌→──────────┐ │0 1 0 1 0 1│ └~──────────┘
'Banana' 'Apple' 'Candy' = 'a'
┌→──────────────────────────────────────┐ │ ┌→──────────┐ ┌→────────┐ ┌→────────┐ │ │ │0 1 0 1 0 1│ │0 0 0 0 0│ │0 1 0 0 0│ │ │ └~──────────┘ └~────────┘ └~────────┘ │ └∊──────────────────────────────────────┘
≠ (Not equal)¶≠ (Unique Mask)¶Monadic ≠ returns 1 on the first occurrence of an item in an array.
≠22 10 22 22 21 10 5 10
┌→──────────────┐ │1 1 0 0 1 0 1 0│ └~──────────────┘
≠ 'Banana'
┌→──────────┐ │1 1 1 0 0 0│ └~──────────┘
≠ 'Mississippi'
┌→────────────────────┐ │1 1 1 0 0 0 0 0 1 0 0│ └~────────────────────┘
≠ (Not Equal To)¶Dyadic ≠ returns true (1) if elements are not equal, and false (0) if elements are equal.
1 2 3 ≠ 4 2 ¯1
┌→────┐ │1 0 1│ └~────┘
The number 7 and the character 7 are not equal.
7 ≠ '7'
1
< (Less than sign)¶< (Less than)¶1 2 3 < 4 2 ¯1
┌→────┐ │1 0 0│ └~────┘
1 2 3 < 2
┌→────┐ │1 0 0│ └~────┘
> (Greater than sign)¶> (Greater than)¶1 2 3 > 4 2 ¯1
┌→────┐ │0 0 1│ └~────┘
1 2 3 > 2
┌→────┐ │0 0 1│ └~────┘
≤ (Less than or equal to sign)¶≤ (Less than or equal to)¶1 2 3 ≤ 4 2 ¯1
┌→────┐ │1 1 0│ └~────┘
1 2 3 ≤ 2
┌→────┐ │1 1 0│ └~────┘
≥ (Greater than or equal to sign)¶≥ (Greater than or equal to)¶1 2 3 ≥ 4 2 ¯1
┌→────┐ │0 1 1│ └~────┘
1 2 3 ≥ 2
┌→────┐ │0 1 1│ └~────┘
≡ (Equal underbar)¶≡ (Depth)¶Depth shows how nested an array is. A simple scalar is an array with no nesting.
≡ 7
0
An array of simple scalars has depth 1.
≡ 'abc'
1
An array that is an array of simple arrays has depth 2.
(1 2)(3 4)
┌→────────────┐ │ ┌→──┐ ┌→──┐ │ │ │1 2│ │3 4│ │ │ └~──┘ └~──┘ │ └∊────────────┘
≡ (1 2)(3 4)
2
If the depth is not consistent, then it returns the max depth as a negative number.
(1 2)(3 4) (5 (6 7))
┌→────────────────────────┐ │ ┌→──┐ ┌→──┐ ┌→────────┐ │ │ │1 2│ │3 4│ │ ┌→──┐ │ │ │ └~──┘ └~──┘ │ 5 │6 7│ │ │ │ │ └~──┘ │ │ │ └∊────────┘ │ └∊────────────────────────┘
≡ (1 2)(3 4) (5 (6 7))
¯3
≡ (Match)¶Match does a comparison to see if 2 objects are equal. It is similar to =, but it works on entire objects rather than elementwise. In other words it is equal with a wider scope.
1 ≡ 1
1
1 ≡ 0
0
≡ works on whole objects and not elementwise with broadcasting.
1 ≡ 1 1
0
≢ (Equal Underbar Slash)¶≢ (Tally)¶Tally counts the major cells in an array. This means it gives the length of the leading axis. For a vector, or rank 1 array, this is the same as it's shape but as a scalar.
≢ 1 2 3
3
⍴ 1 2 3
┌→┐ │3│ └~┘
If there are multiple dimensions, Tally returns the number of major cells, which is the size of the first dimension
≢ 2 3 ⍴ ⍳6
2
≢ 3 2 ⍴ ⍳6
3
≢ (Not match)¶Not match does a comparison to see if 2 objects are not equal. It is similar to ≠, but it works on entire objects rather than elementwise. In other words, it is equal with a wider scope.
1 ≢ 1
0
1 ≢ 0
1
≢ works on whole objects and not elementwise with broadcasting.
1 ≢ 1 1
1
∨ (Logical or)¶∨ (Greatest Common Divisor (Or))¶Standard or operator when applied to booleans.
0 1 0 1 ∨ 0 0 1 1 ⍝ Truth table for *or*
┌→──────┐ │0 1 1 1│ └~──────┘
5 is the largest number that divides 15 and 35 cleanly (meaning no remainder). Therefore, 5 is the greatest common divisor of 15 and 35.
15 1 2 7 ∨ 35 1 4 0 ⍝ GCD
┌→──────┐ │5 1 2 7│ └~──────┘
⍱ (Logical NOR)¶⍱ (Nor)¶For more details on logical nor, see this page
0 1 0 1 ⍱ 0 0 1 1 ⍝ Truth table for *nor*
┌→──────┐ │1 0 0 0│ └~──────┘
∧ (Logical AND)¶∧ (Lowest Common Multiple (And))¶Standard and operator when applied to booleans.
0 1 0 1 ∧ 0 0 1 1 ⍝ Truth table for *and*
┌→──────┐ │0 0 0 1│ └~──────┘
105 is the smallest multiple of both 15 and 35, therefore 105 is the lowest common multiple of 15 and 35.
15 1 2 7 ∧ 35 1 4 0 ⍝ LCM
┌→────────┐ │105 1 4 0│ └~────────┘
~ (Logical NOT)¶~ (Not)¶~ 0 1 ⍝ Truth table for *not*
┌→──┐ │1 0│ └~──┘
~ (Without;Excluding)¶Gives the elements from the left argument that are not in the right.
3 1 4 1 5 ~ 5 1
┌→──┐ │3 4│ └~──┘
Also works on the character vectors
'aa' 'bb' 'cc' 'bb' ~ 'bb' 'xx'
┌→──────────┐ │ ┌→─┐ ┌→─┐ │ │ │aa│ │cc│ │ │ └──┘ └──┘ │ └∊──────────┘
Also works on nested arrays
(1 2) 3 ~ ⊂1 2
┌→┐ │3│ └~┘
⍲ (Logical NAND)¶⍲ (Nand)¶Nand is "Not and"
0 1 0 1 ⍲ 0 0 1 1 ⍝ Truth table for *nand*
┌→──────┐ │1 1 1 0│ └~──────┘
We could use and (^) apply not (~) afterward to get the same result. As you can see it truly is "Not and"
~ 0 1 0 1 ∧ 0 0 1 1 ⍝ Truth table for *nand*
┌→──────┐ │1 1 1 0│ └~──────┘
/ (Slash)¶/ (Replicate)¶v←22 10 22 22 21 10 5 10
≠v
┌→──────────────┐ │1 1 0 0 1 0 1 0│ └~──────────────┘
(≠v)/v
┌→─────────┐ │22 10 21 5│ └~─────────┘
3 1 3 1 3 1 / 'Banana'
┌→───────────┐ │BBBannnannna│ └────────────┘
⍳ (iota)¶⍳ (index generator)¶Creates index locations for array with specified shape.
⍳4
┌→──────┐ │1 2 3 4│ └~──────┘
⍳2 3
┌→──────────────────┐ ↓ ┌→──┐ ┌→──┐ ┌→──┐ │ │ │1 1│ │1 2│ │1 3│ │ │ └~──┘ └~──┘ └~──┘ │ │ ┌→──┐ ┌→──┐ ┌→──┐ │ │ │2 1│ │2 2│ │2 3│ │ │ └~──┘ └~──┘ └~──┘ │ └∊──────────────────┘
⍳ is often used with ⍴ to create arrays
2 3 ⍴ ⍳6
┌→────┐ ↓1 2 3│ │4 5 6│ └~────┘
To generate an array with various patterns you can combine with other functions.
1+2×⍳6
┌→────────────┐ │3 5 7 9 11 13│ └~────────────┘
Array of shape 0 is nan empty vector and the index locations of a shape 0 array is and empty vector.
⍳0
┌⊖┐ │0│ └~┘
⍳ (index of)¶Provides the index locations for where ⍵ (right argument) is in ⍺ (left argument)
1 3 6 5 4 ⍳ 3
2
'ABCDABCDEF' ⍳ 'ACFG'
┌→────────┐ │1 3 10 11│ └~────────┘
⎕←mat←3 2 ⍴ ⍳6
┌→──┐ ↓1 2│ │3 4│ │5 6│ └~──┘
⍳ works on major cells, or along the first dimension. The third major cell of mat is 5 6.
mat←3 2 ⍴ ⍳6
mat ⍳ 5 6
3
⍸ (iota underbar)¶⍸ (Where)¶When applied to a boolean/binary array, where gives index locations of the true values.
⍸ 1 0 0 1 1
┌→────┐ │1 4 5│ └~────┘
When applied to a natural number, where repeats the index locations if the number is greater than 1. For example ⍸1 2 3 would give 1 2 2 3 3 3
⍸1 2 3
┌→──────────┐ │1 2 2 3 3 3│ └~──────────┘
⍸ 2 0 0 2 1
┌→────────┐ │1 1 4 4 5│ └~────────┘
When applied to a higher rank array it still provides the index locations of the elements, but index locations have multiple values (in this case row and column).
⎕←bmat ← 2 3 ⍴ 0 1 0 1 0 1
⍸ bmat
┌→────┐ ↓0 1 0│ │1 0 1│ └~────┘
┌→──────────────────┐ │ ┌→──┐ ┌→──┐ ┌→──┐ │ │ │1 2│ │2 1│ │2 3│ │ │ └~──┘ └~──┘ └~──┘ │ └∊──────────────────┘
⎕←bmat ← 2 3 ⍴ 0 2 0 2 0 2
⍸ bmat
┌→────┐ ↓0 2 0│ │2 0 2│ └~────┘
┌→────────────────────────────────────┐ │ ┌→──┐ ┌→──┐ ┌→──┐ ┌→──┐ ┌→──┐ ┌→──┐ │ │ │1 2│ │1 2│ │2 1│ │2 1│ │2 3│ │2 3│ │ │ └~──┘ └~──┘ └~──┘ └~──┘ └~──┘ └~──┘ │ └∊────────────────────────────────────┘
⍸ (Interval Index)¶Interval index creates intervals and returns an array that tells you which interval a value falls in.
If the following example, ⍵ is 2 4 6. Therefore the following intervals (sometimes called bins) are created.
⍺ (right argument) is 1 2 3 4 5 6 7 so let's put those into intervals, which the first interval starting at 0.
⍺ is 1. 1 is Less than 2. Therefore it is in the 0th interval and interval index returns 0.⍺ is 2. 2 is in the 2-4 range. Therefore it is in the 1st interval and interval index returns 1.⍺ is 3. 3 is in the 2-4 range. Therefore it is in the 1st interval and interval index returns 1.⍺ is 4. 4 is in the 4-6 range. Therefore it is in the 2nd interval and interval index returns 2.And so on and so forth.
2 4 6 ⍸ 1 2 3 4 5 6 7
┌→────────────┐ │0 1 1 2 2 3 3│ └~────────────┘
Interval index works the same way with character vectors.
D is in the first interval because it is between A and E.Y is in the last interval (fifth) because it is larger than all valuesA is in the first interval because it is between A and E.L is in the third interval because it is between I and O.etc.
'AEIOU' ⍸ 'DYALOG'
┌→──────────┐ │1 5 1 3 4 2│ └~──────────┘
Interval index works across major cells.
⎕←mat←3 2⍴⍳6
┌→──┐ ↓1 2│ │3 4│ │5 6│ └~──┘
mat ⍸ 3 3
1
mat ⍸ 3 5
2
mat ⍸ 2 2 ⍴ 3 3 3 5
┌→──┐ │1 2│ └~──┘
⌈ (Upstile)¶⌈ (Ceiling)¶Rounds up to nearest whole number.
⌈ 3.4 ¯3.4 3 0
┌→───────┐ │4 ¯3 3 0│ └~───────┘
⌈ (Maximum)¶Takes maximum of 2 values
3⌈2
3
3 2⌈2 3
┌→──┐ │3 3│ └~──┘
4 ⌈ 6 ⌈ 2
6
a ← ¯4 6 2
0 ⌈ a
┌→────┐ │0 6 2│ └~────┘
⌊ (Downstile)¶⌊ (Floor)¶Rounds down to nearest whole number
⌊ 3.4 ¯3.4 3 0
┌→───────┐ │3 ¯4 3 0│ └~───────┘
⌊ (Minimum)¶Takes minimum of 2 values
4 ⌊ 6 ⌊ 2.5
2.5