Relational and logical functions

This part will cover

Functions used to compare values
Comparing vectors and scalars
Functions used to perform bitwise operations
Bitwise operations on scalars and vectors

APL also comes the standard comparison functions, with the expected representative symbols.

Dyadic =, ≠, ≤, <, >, ≥ : Comparison Functions

      0 = 0
1
      0 ≠ 0
0
      3 ≤ 3
1
      3 < 3
0
      ⍝ Getting a mask for numbers which divide 2
      0 = 2 | 1 2 3 4 5 6 7 8 9 10
0 1 0 1 0 1 0 1 0 1

Note that when you use these functions over two vectors, the result is a vector, denoting the function element wise. To compare arrays, use the dyadic ≡ match function.

      1 2 1 4 5 2 5 2 3 2 = 1 2 3 4 5 2 3 2 4 2
1 1 0 1 1 1 0 1 0 1
      1 2 1 4 5 2 5 2 3 2 ≤ 1 2 3 4 5 2 3 2 4 2
1 1 1 1 1 1 0 1 1 1
      1 2 1 4 5 2 5 2 3 2 ≡ 1 2 3 4 5 2 3 2 4 2
0
      1 2 1 4 5 2 5 2 3 2 ≡ 1 2 1 4 5 2 5 2 3 2
1

      'Hello' ≡ 'hello'
0
      'hello' ≡ 'hello'
1

One useful feature of APL in real-world applications is the ability to change the tolerance of comparison when it comes to floating point values (except for comparison against zero!), and the displayed precision of numbers. The tolerance can be read (and set) via the ⎕CT system variable, and the precision shown is read (and set) via ⎕PP. The tolerance can be anything from 0 (exact comparisons) to 10*¯10.

      ⍝ 1 + Sine of Pi
      X ← 1 + 1 ○ ○ 1
      X
1

      ⎕PP ← 20
      X
1.0000000000000002

      ⎕CT ← 0
      1 = X
0

      ⎕CT ← 1E¯10
      1 = X
1

An amusing quote from the APL Wiki about comparison tolerance

In an early talk Ken was explaining the advantages of tolerant comparison. A member of the audience asked incredulously, "Surely you don't mean that when A=B and B=C, A may not equal C?" Without skipping a beat, Ken replied, "Any carpenter knows that!" and went on to the next question.

—Paul Berry

As suggested by the choice of glyphs for the Greatest Common Divisor ∧ and the Least Common Multiple ∨, these two operations are also used for the boolean logic "or" and "and" operations. This is because 0 is divisible by everything, as the remainder is always 0. So the Greatest Common Divisor of 0 and x is just x, since x divides both 0 and x. Dually, 0 is also a multiple of everything, so the Least Common Multiple of 0 and x is just 0, since 0 is a multiple of both x and 0 (and it’s the smallest!).

      3 ∧ 0
0
      0 ∨ 3
3
      ⍝ Checking for numbers which divide 3
      triples ← 0 = 3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
      triples
0 0 1 0 0 1 0 0 1 0 0 1 0 0 1

      ⍝ Checking for numbers which divide 5
      quintuples ← 0 = 5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
      quintuples
0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

      ⍝ Checking for numbers which divide 3 and 5
      triples ∧ quintuples
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

      ⍝ Checking for numbers which divide 3 or 5
      triples ∨ quintuples
0 0 1 0 1 1 0 0 1 1 0 1 0 0 1

Monadic ~ : Logical Not Dyadic ⍲ & ⍱ : Logical Nand and Logical Nor

      ~ 1 1 0 1 1
0 0 1 0 0

      ⍝ However, this operator is truly logical
      ⍝ You cannot take the not of a non-0 non-1 number
      ~ 3 
DOMAIN ERROR
      ~3
      ∧

      ⍝ Numbers which divide anything other than both 3 and 5
      triples ⍲ quintuples
1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

      ⍝ Numbers which divide anything other than either 3 or 5
      1 1 0 1 1 ⍱ 1 0 1 0 1 
1 1 0 1 0 0 1 1 0 0 1 0 1 1 0