Sets in Practical Situations

## Sets in Practical Situations

If there are a large number of sequential members in a set, write as A={ 1,3,5,7, ...} (all odd numbers, to infinity). The ellipses (...) means continue the sequence. If there is a number following the ellipses, then the sequence ends at that number: B={1,3,5,7,...21} is a set of odd numbers from 1 to 21.

A colon in a pair of curly brackets is used to identify elements in a set. For example, {n : n>7} means values of n such that the elements of the set are greater than 7.

Sets can be used to define a set of points eg C = {(x, y) : y = 3x2 + 4} which defines a set of coordinates that satisfies the given equation.

There are four groups of numbers that are often used in sets.

is the set of natural, positive numbers. If the set includes zero, it is normally shown as ℕ0; if the set does not include zero, then it should be shown as ℕ1.

is the set of all integers.

is the set of Rational numbers (numbers that can be written as a fraction, where the numerator and denominator of the fraction are both integers.

is the set of real numbers, and includes π, √2 etc.

## Example 1

Given a set K = {n ∈ ℕ: n≤10} and a set L = {3,4,5,...20}, what is n(K ∩ L)?

The set K is the set of natural numbers that are less than or equal to 10: K = {1,2,3,4,5,6,7,8,9,10}.

L= {3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}.

Elements in both sets K ∩ L = {3,4,5,6,7,8,9,10}.

The number of elements is 8.

Answer: n(K ∩ L) = 8

## Example 2

Given ε = {1, 2, 3, … 20}

Set A is a list of prime numbers from 2 to 20

Set B is defined as {n: 0 < n < 7}

What is the probability P(A ∩ B)?

A = {2, 3, 5, 7, 11, 13, 17, 19}

B = {1, 2, 3, 4, 5, 6}

A ∩ B = {2, 3, 5}, and n(A ∩ B) = 3

The Universal Set has 20 elements, so probability is frac(3)(20) or 0.15