Difference between ‘..’ (double-dot) and ‘…’ (triple-dot) in range generation?

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The documentation for Range† says this:

Ranges constructed using .. run from the beginning to the end inclusively. Those created using ... exclude the end value.

So a..b is like a <= x <= b, whereas a...b is like a <= x < b.

Note that, while to_a on a Range of integers gives a collection of integers, a Range is not a set of values, but simply a pair of start/end values:

(1..5).include?(5)           #=> true
(1...5).include?(5)          #=> false

(1..4).include?(4.1)         #=> false
(1...5).include?(4.1)        #=> true
(1..4).to_a == (1...5).to_a  #=> true
(1..4) == (1...5)            #=> false

†The docs used to not include this, instead requiring reading the Pickaxe’s section on Ranges. Thanks to @MarkAmery (see below) for noting this update.

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