In this topic of IBDP Mathematics, we will be discussing the concept of sets, and different set notation. This topic is in the core part of the HL and SL syllabus, so is required for all IB Maths students.
Sets
In IBDP Mathematics, sets are defined as a collection of distinct numbers or objects. Each object/number within a set is known as an element or member of the set.
- For example, we can define the colours of the rainbow are a set. So, the colours 'red', 'orange', 'yellow', 'green', 'blue', 'violet' and 'indigo' form a set.
- We can also say that each individual colour is an element of the set - e.g. 'red' is an element of the set.
Each element in the set only needs to be written once - there should be no duplicated elements. When we are writing down sets, we write its elements within curly brackets, with commas in between each member. The set is also represented using a capital letter, making it easier to refer to the set.
- In the example above, we can represent the set with a capital C.
- Therefore, the set can be written out as such - C = {colours of the rainbow} = {red, orange, yellow, green, blue, violet, indigo}
When writing out sets, there are several types of set notation to allow us to describe and talk about different sets. The set notations is defined as follows:
- ∈ means 'is an element of' or 'is in'
- ∉ means 'is not an element of' or 'is not in'
- n(A) means 'the number of elements in set A'
For example, say E = {even numbers less than or equal to 10} = {2,4,6,8,10}. We can say that:
- 4 ∈ E - i.e. 4 is an element of E
- 5 ∉ E - i.e. 5 is not an element of E
- n(E) = 5, as there are 5 elements in set E.
In IBDP Mathematics, there are also some set definitions you need to know:
- Two sets are equal if they contain exactly the same elements.
- Set A is a subset of set B if every element of A is also an element of B - in other words, for each element in set A, it is also an element in set B.
- We can write this as A ⊆ B - this can be read as 'A is a subset of B'
- For example, say A = {1,2,3} and B = {1,2,3,4,5}. We can say that A ⊆ B, as the elements in set A are also elements of set B. Thus, A is a subset of B.
- It is important to note that it is possible for set A and B to be equal, even if A is a subset of B. If A and B are equal, then every element of A is also an element of B - this is precisely the definition of A ⊆ B.
- A is a proper subset of B if every element of A is also an element of B, but A and B are not equal.
- We can write this as A ⊂ B.
- The example above (where A = {1,2,3} and B = {1,2,3,4,5}) is also an example of a proper subset - A is a proper subset of B, as all the elements of A are also elements of B, but A is not equal to B.
- The empty set ∅ or { } is a set which contains no elements.
- Note: The empty set is a subset of all sets.
- Set A is considered a finite set if n(A) has a particularly defined value.
- A set is a infinite set if it has an endless number of elements.
Intersection and Union
In IBDP mathematics, we often are dealing with more than one set. With more than one set, we will often have to look at the intersection or union of the two sets - this information helps us describe and understand the relationship between the two sets.
The intersection of two sets A and B is the set of elements that are both in set A and set B.
- We can write this as A ⋂ B - this can be read as 'the intersection of A and B'
- For example, say A = {2,3,4,7} and B = {2,4,6,8}. We can thus write A ⋂ B = {2,4}, since 2 and 4 are elements of both sets.
The intersection gives us information as to how much overlap there is between the two sets. The larger the number of elements in the intersection of two sets, the more overlap there is between the sets. We can also use the intersection to determine if a set is the subset of another set.
- Say A = {1,2,3} and B = {1,2,3,4,5}. Thus, A ⋂ B = {1,2,3}. However, this is precisely set A, thus we can deduce that A is the subset of B.
If there is no intersection between two sets (i.e. there are no elements in common), then the sets are described as disjoint or mutually exclusive.
- In this case, A ⋂ B = ∅ - it contains no elements.
- An example of disjointed/mutually exclusive sets would be if A = {1,3,5,7} and B = {2,4,6,8}.
The union of two sets A and B is the set of elements that are either in A or B.
- We can write this as A ⋃ B - this can be read as 'the union of A and B'
- For example, say A = {2,3,4,7} and B = {2,4,6,8}. We can thus write A ⋃ B = {2,3,4,6,7,8}.
- It is important to note that although the elements 2 and 4 appear in both sets, we only need to write them once - there is no need to duplicate the elements.
- A good way to write out the union of two sets would be to first copy down the elements in one set, then write down the elements of the other set, excluding numbers which are already written.
Complement of a set
In IBDP Mathematics, we define the universal set U as the set of all the elements we are considering.
- For example, if we are considering the digits we write whole numbers with, the universal set is U = {0,1,2,3,4,5,6,7,8,9}
The complement of a set is the set of all elements of U that are not elements of set A.
- We can write this as A' - this can be read as 'the complement of set A'
- For example, consider U = {0,1,2,3,4,5,6,7,8,9} and A = {1,2,4,5,6}. Thus, A' = {0,3,7,8,9} as these are the elements within the universal set that are not in set A.
There are 3 things that can be observed about complementary sets:
- A ⋂ A' = ∅ - the complementary sets are mutually exclusive, as A and A' have no common elements.
- This is fairly intuitive, as the complement of set A contains the elements that aren't contained within set A, thus there should be no common elements.
- A ⋃ A' = U - as all elements in A and A' combined make up U.
- mathsThis should also be fairly intuitve, as A' consists of the elements in the universal set that was not in A. Thus, the elements of set A and A' combined should form the universal set.
- n(A) + n(A') = n(U) - in other words, the sum of the number of elements in set A and A' should equal the number of elements in the universal set.
This is the end of this topic.
