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interval


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A finite interval on the real line is a subset of R defined in terms of end-points a and b. Since each end-point may or may not belong to the subset, there are four types of finite interval: (i) the closed interval {x | xR and axb}, denoted by [a, b],(ii) the open interval {x | xR and a < x < b}, denoted by (a, b),(iii) the interval {x | xR and ax <b}, denoted by [a, b),(iv) the interval {x | xR and a < xb}, denoted by (a, b].There are also four types of infinite interval:(v) {x | xR and ax}, denoted by [a, ∞),(vi) {x | xR and a < x}, denoted by (a, ∞),(vii) {x | xR and xa}, denoted by (−∞, a],(viii) {x | xR and x < a}, denoted by (∞, a).Here ∞ (read as ‘infinity’) and −∞ (read as ‘minus infinity’) are not, of course, real numbers, but the use of these symbols provides a convenient notation.

(i) the closed interval {x | xR and axb}, denoted by [a, b],

(ii) the open interval {x | xR and a < x < b}, denoted by (a, b),

(iii) the interval {x | xR and ax <b}, denoted by [a, b),

(iv) the interval {x | xR and a < xb}, denoted by (a, b].

(v) {x | xR and ax}, denoted by [a, ∞),

(vi) {x | xR and a < x}, denoted by (a, ∞),

(vii) {x | xR and xa}, denoted by (−∞, a],

(viii) {x | xR and x < a}, denoted by (∞, a).

If I is any of the intervals (i) to (iv), the open interval determined by I is (a, b); if I is (v) or (vi), it is (a, ∞) and, if I is (vii) or (viii), it is (−∞, a).

Subjects: Mathematics.


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