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What are the most important definitions for set theory and notation?
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Set: a collection of numbers; each number is called an element of the set:

ℝ, ℤ; {−1; 0; 1}, {1; __ 1 2 ; __ 1 4 ; __ 1 8 } etc. are all sets.

Element: any one of the numbers in a set; we show that a number is an element of a set by writing 2 ∈ ℤ and that it is not, by writing √ _ 2 ∉ ℤ (we say that 2 ‘is an element of’ the integers and that $\sqrt{2}$ ‘is not an element of’ the integers).

Cardinality: the number of elements in a set; all the sets in our hierarchy have an infinite cardinality except the set for zero, which we can represent as {0} and which has a cardinality of 1; the sets in our example above has a cardinality of 6. The set of rational numbers between 0 and 1 are infinite because there is no limit to how many you can count.

Subset: a set that has some of the elements of another set in it, for instance, the integers are a subset of the rational numbers and the set in our example is a subset of the natural numbers. We have seen that we can write these as ℤ ⊆ ℚ and {1; 2; 3; 4; 5; 6} ⊆ ℕ.

Interval: a subset of all the numbers between two given numbers called the end points. The set in our example is an interval in the integers.

There is special notation for intervals: {x ∈ ℤ | 1 ≤ x ≤ 6}.

This is read as: {……} meaning ‘the set of …’ x ∈ ℤ meaning ‘… all elements x in the set of integers …’ | meaning ‘… such that…’ 1 ≤ x ≤ 6 meaning ‘… x is between and including 1 to 6’ The set in our example could also be written as {x ∈ ℤ | 0 < x < 7} where 0 < x < 7 means all the numbers between but not including 0 and 7.

Interval notation: the short version of this is to write x ∈ [1; 6] for the first and x ∈ (0; 7) for the second representation. [ and ] means include the end points of the interval while ( and ) mean exclude the end points.

by Diamond (75,918 points)

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