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An unordered pair is a finite set; its cardinality (number of elements) is 2 or (if the two elements are not distinct) 1. In axiomatic set theory, the existence of unordered pairs is required by an axiom, the axiom of pairing. More generally, an unordered n-tuple is a set of the form {a 1, a 2,... a n}. [5] [6] [7]
The existence of a contrastive distribution between two speech sound plays an important role in establishing that they belong to two separate phonemes in a given language. [ 1 ] For example, in English , the speech sounds [pʰ] and [b̥] can both occur at the beginning of a word, as in the words pat and bat .
Many languages show contrasts between long and short vowels and consonants. A distinctive difference in length is attributed by some phonologists to a unit called a chroneme. Thus, Italian has the following minimal pair that is based on long and short /l/:
Another example in English of a phonemic contrast would be the difference between leak and league; the minimal difference of voicing between [k] and [g] does lead to the two utterances being perceived as different words. On the other hand, an example that is not a phonemic contrast in English is the difference between [sit] and [siːt]. [1]
A paired difference test is designed for situations where there is dependence between pairs of measurements (in which case a test designed for comparing two independent samples would not be appropriate). That applies in a within-subjects study design, i.e., in a study where the same set of subjects undergo both of the conditions being compared.
As (,) quantifies the number of pairs with unequal X values, Somers’ D is the difference between the number of concordant and discordant pairs, divided by the number of pairs with X values in the pair being unequal.
The set of all ordered pairs whose first entry is in some set A and whose second entry is in some set B is called the Cartesian product of A and B, and written A × B. A binary relation between sets A and B is a subset of A × B. The (a, b) notation may be used for other purposes, most notably as denoting open intervals on the real number line ...
The Kendall tau distance between two series is the total number of discordant pairs. The Kendall tau rank correlation coefficient, which measures how closely related two series of numbers are, is proportional to the difference between the number of concordant pairs and the number of discordant pairs.