Sets

Sets

GCSE(F),

Sets are collections of data. Values that belong to a set are shown inside curly brackets { }.

A set might not contain numbers. It might contain items, such as the colours of a flag {red, white, blue}. Values in a set are known as elements. 3{odd numbers} means that 3 is an element of the set of odd numbers. 2{odd numbers} means that 2 does not belong to that set. The number of members in set A is given by n(A).

A set that contains all possible elements for a given situation is a Universal Set, and is written as `xi`. A universal set that contains all the scores that can be thrown with two dice is:

`xi = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}`

If set O contained all odd numbers that can be thrown with two dice, and set D contained all odd numbers with a single digit, then Set D is a subset of O, as all the elements in D are also in set O. This is written as DO.

Set Oc (sometimes O`) are all the elements that are not in O: this is the complement of set O. Oc={2,4,6,8,10,12}

A colon in a pair of curly brackets is used to select items from a set. Using the two dice example, {n∈O:n>7} selects the elements that belong to set O (the n∈O part) that are higher than 7 (the n>7 part). Therefore {n∈O:n>7} ={9,11}

An empty set is shown with the sign ∅, and represents the set {}.

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.

Examples

1. The number cards in a suit of playing cards are defined as the universal set, such that
`xi = {2, 3, 4, 5, 6, 7, 8, 9 and 10}`.

Set E contains all the even numbers. Write out the members of that set.

Answer: E = {2, 4, 6, 8, 10}

Only the even numbers are required. Remember that the curly brackets are required.

2. The number cards in a suit of playing cards are defined as the universal set, so
`xi = {2, 3, 4, 5, 6, 7, 8, 9 and 10}`.

If E contains the set of even numbers, write the members of Ec.

Answer: Ec = {3, 5, 7, 9}

The complement of E is required: those elements that are not in E. Use a small c to show it is a complement.