DESCRIPTION OF SETS

SETS

Hello lovely students, welcome to another episode of learning. Today we are looking at sets.

A set is collection of items that are related. They are represented with a block letter and the members placed in curly brackets. Eg. A = {Ama, Afia, Yaa}.

Representation of Sets.

A set can be represented in three (3) basic forms. These are

1.      Listing the members

This is done by representing a set with a block letter and listing the members in a curly bracket with comma (,) separating each member from the other. Examples are giving below.

A = {2, 4, 6, 8}, X = {Rice, maize, millet}.

In the first set, A is the name of the set and the members are 2, 4, 6 and 8. In set X, the members are rice, maize and millet.

2.      Describing the set.

This is done by representing the set with a block letter with the description in a curly bracket.

From the first type. A can be written as A = {even numbers less than 10} whiles X = {examples of cereals}

3.      Builder Notations.

This is done with the less than signs with a variable in between. It is then followed by description of the variable. Example A = {2 ≤ x ≤ 8, where x is an even number}, B = {3 < y < 6, where y is an integer}

NB: In most cases when the second and third types are given, you’ll be required to change them into the first type.

Let’s solve these questions

List the members of the following sets.

a.       B = {prime numbers less than 10}

b.      Y = { -5 < x < 0, where x is an integer}

c.       M = {-2 ≤ a ≤ 4, where a is an even integer}

Solutions

a.       Prime numbers less than 10 are listed as 2,3,5 and 7. This can be represented as

B = {2, 3, 5, 7}. Note that 2 is a prime number and 9 is not

b.      For Y, we need integers that are more than -5 but less than 0. This means we start from -4 and end at -1. i.e. Y = {-4, -3, -2, -1}. Note that -4 and 0 are not part since the sign is less than (<) and not less than or equal to (≤)

c.       For M, we are writing even integers from -2 to 4. This means we start from -2 and end at 4 because of the sign less than or equal to (≤). i.e. M = {-2, 0, 2, 4}.

Now try your hand on these by listing the members of the following sets.

1.      X = {names of the days of the week}

2.      Y = {week day names of Girls in Akan}

3.      P = {-4 < a < 11, where a is an integer}

4.      N = {6 ≤ b ≤ 12, where b is an odd number}

5.      T = {Factors of 36}.

Types of sets.

Now let us look at some types of sets.

1.      An empty (Null) set – This is a set with no member in it. It is denoted as {}. For example If P = {even factors of 3}, then P is an empty set since there are no factors of 3 that are even numbers.

2.      A unit set – This is a set with only one member. For example X = {even prime numbers}, since 2 is the only even number that is also prime, we can solve this as X = {2}. i.e. X is a unit set.

3.      Universal set – This is a set where we derive other sets from. A universal set can be described as a mother set which we can get daughter sets (subsets) from.

4.      A subset – is a smaller set derived from a bigger set. For example Given that A = { 1,2,3}, we can derive eight(8) subsets from A. i.e. {}, {1}, {2}, {3}, {1,2} {2,3}, {1,3}, {1,2,3}. These are the subsets of set A. We can also be tasked to just find the number of subsets we can derive from a set. This is obtained by the formula 2ⁿ, where n is the number of elements (members) of the set.

So if P = {a, b, c, d}, there are for (4) members in the set so the number of subsets is given as 2⁴ = 16.

5.      Union of Sets (U) – This is obtained by combining all the elements in two sets without repeating the common members. Example if A = {1,2,4} and B {1, 3,4}. Then A U B = {1,2,3,4}

6.      Intersection of sets (∩) – Two sets that have common members are said to have an intersection. For example if A = {multiples of 4 less than 25} and B = {Factors of 60}. By listen the members

A = {4, 8, 12, 16, 20, 24} and B = {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}. We can see that the numbers 4, 12 and 20 can be found in both sets. So A∩B = {4, 12, 20}.

7.      Disjoint sets – These are two sets with no members in common. For example if A = {multiples of 4 less than 25} and C = {factors of 30}, since factors of 30 are 1,2,3,5,6,10,15,and 30, there are no known members in common so we can say that A∩C = {} or A and C are disjoint sets.

8.      Complements of sets – For this part, we need to revisit universal sets and subsets. If A is a subset of U such that U = {whole numbers up to 10} and A = {even numbers}. Try and list the members of each set. NB. A is a subset of U so it should not pass 10.

The members are U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}. Now list the members of U that are not found in A. These are 0, 1, 3, 5, 7 and 9. These are known as complement of A. It is represented as A¹ = {0, 1, 3, 5, 7, 9}.

9.      Equal sets – Two sets are said to be equal if they have the same members. For example if A = {1,2,3,4} and B = {4, 2, 1, 3} then A is equal to B. so A = B.

10.  Equivalent sets – If two sets have the same number of members, they are said to be equivalent. For example A = {1,2,3,4} and B = {5,6,7,8}. Each set has four members. This can be written as n(A) = 4 and n(B) = 4. Therefore A and B are equivalent. This is written as n(A) = n(B).

Do not confuse the two.

Now that we have seen all the types and properties of sets. Let us solve this question.

If X = { - 2 < b < 5, where b is an odd integer} and Y = { positive factors of 6} and U = {-3, -2, -1, …………….. 8}. Given that X and Y are subsets of U,

a.       List the members of

i.                    U, X and Y

ii.                  X¹

iii.                Y¹

iv.                X∩Y

v.                  X U Y.

b.      Find the following

i.                    X¹ ∩ Y

ii.                  X ∩ Y¹

iii.                X¹ ∩ Y¹

iv.                (X U Y)^1.

Solution

a.       To list the members of each set, note that X and Y are subsets and can only contain members of U.

i.                    U = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8}

X = {-1, 1, 3, 5, 7}

Y = {1, 2, 3, 6}

ii.                  X¹ = {-3, -2, 0, 2, 4, 6, 8}

iii.                Y¹ = {-3, -2, -1, 0, 4, 5, 7, 8}

iv.                X∩Y = {1, 3}

v.                  X U Y = {-1, 1, 2, 3, 5, 6, 7}

b.      We can solve these by using the answers we got from above.

i.                    X¹ ∩ Y = {2, 6}

ii.                  X ∩ Y¹ = {-1, 5, 7}

iii.                X¹ ∩ Y¹ = {-3, -2, 0, 4, 8}

iv.                (X U Y)¹ = {-3, -2, 0, 4, 8]

Now try and solve these questions on your own.

1.      Find the members of the set A = {-3 < x < 4, where x is an integer}

2.      Given that A = {Months of the year ending with letter X} what type of set is A?

3.      Given that P = {a. c, e, f, t}, how many subsets can be obtained from P?

4.      Find n(Y) if Y = {4, 6, 8}.

5.      Given that A and B are subsets of the universal set E such that E = {counting numbers up to 9}, A = {multiples of 2} and B = {Factors of 6}.

a.       List the members of E, A and B

b.      Find

i.                    A¹ ∩ B

ii.                  A ∩ B

iii.                A ∩ B¹

iv.                (A ∩ B)¹

v.                  A U B¹.

We will continue this topic next time by introducing Venn diagrams. Don’t forget to share this with your friends and colleagues.

Till we meet again at this same place its bye for now.