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# Sets

## 1.2 Describing a Set and Function

#### Describing a Set

• It is not always possible to list all the elements in a set (e.g. N, Z, Q, R as above). There is a way of describing sets that is useful for infinite sets (i.e. sets with an infinite number of elements).
• The set of all x such that ____ is denoted as \left\{x:_{-----}\right\}. For example:

i) \{x:0<,\ x<1\} is the set of real numbers strictly between 0 and 1.

ii) \{x:x\leq3\} is the set of real numbers smaller or equal to 3.

iii) \{x: x>0, x \in Q\} is the set of all positive rational numbers.

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## 1.1 Set Notations and Sets of Numbers

#### Set Notations

• A set is a collection of objects. The objects that are in the set are known as elements or members of the set.
• If x is an element of a set A, we write x\in A. We read this as ‘x is an element of set A’ or ‘x is in A’.
• If x is not an element of a set A, we write x \notin A.
• The set B is called a subset of a set A if every element of B is also an element of A. We write B ⊆ A. This expression can also be read as ‘B is contained in A’ or ‘A contains B’.
• An example is given at the right using Venn Diagram. Here, A=\{0,1,2,3,4\},\ B=\{0,1,2\}.
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