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## Posted by Chester Morton / Monday 24 April 2017 / No comments

### Understanding the basics of sets

#
**SETS**

# A set is a collection of well-defined objects. For example a Mathematical Set. It contains Mathematical Instruments e.g. a Ruler, Template, a Pair of divider, a Pair of Compass, Sets Square etc. What they have in common is that, they are all used in drawing. Another example is the cutlery set. This includes Knife, Spoon and Fork, which we use on the dining table.

## Ç is a sign of intersection of sets.

È is a sign of unity or union.

{}
or f denotes empty or null set.

Ì is a sign of subset.

É is a sign of super set.

h (A) is a
sign that describes the number of elements of set A.

Î is a sign
of belonging to a set or a member of a set.

Ï is a sign
of not belonging to a set or not a member of a set

U
or E. denotes universal set and it is defined as the totality of all sets under
consideration.

## TYPES OF SET

Equal
Sets: Equal sets as the name implies are two or more sets having the same
elements in them. For instance the set A = {1, 2, 4} is equal to the set B =
{2, 1, 4}.

Disjoint
Sets: Two or more sets are said to be disjoint if they do not have any element
in common. For instance X = {1, 3, 5} and Y = {2, 4, 6} are disjoint sets.

Joint
or Intersection Sets: Two or more sets are said to be joint if at least one
element is common to them or at least one element in one is found in the other.
For instance the set Z = {1, 2, 3, 4} and Y = {1, 3, 5, 6} are joint sets
because they have two elements in common i.e. 1 and 3. Thus 1 and 3 become the
point of intersection of the two sets.

Subset:
If all elements of a set belong to another set, then we can say that the first
set is a subset of the second. e. g. the set A = {1, 2, 3} is a subset of the
set B = {1, 2, 3, 4, 5, 6}.

Note
that an empty set is a subset of every set and a set is a subset of itself i.e.
the set

Y
= {2, 4, 6} is a subset of itself.

To
get the number of subsets of a set, which has n elements, we use this formula 2

^{n}.
For
example a set of 2 elements will have 2

^{2}i.e. 2 x 2 = 4 subsets.
Example1.1 If U = {0, 1, 2}, list all the subsets of
U. There are 3 elements in the set hence
the number of subset = 2

^{n}, where n is the number of elements in the set i.e.
n
= 3. So we have 2

^{3}= 2 x 2 x 2 = 8 i.e. there are 8 subsets. They are { }, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2} and {0, 1, 2}.
Null
or Empty Set: Any set that has no
element in it is a null set or an empty set and it is denoted by the sign f or {}, Empty set is
a subset of itself and any other set.

Complementary
Set: This is normally defined with respect to a universal set. If the set A is
a subset of a universal set U, then all the elements in the universal set which
can not be found in the set A become the complementary set of A which is
denoted A

^{/}e.g. if X = {2, 4, 6} is a subset of U = {1, 2, 3, 4, 5, 6} then the set X^{/}= {1, 3, 5}.
Example1.2 The
universal set U = {2, 3, 5, 7}, P = {2, 5} and Q = {5, 7},

Find: i. (PÇQ)

^{ /}ii. P^{/ }È Q^{/}
iii. State
the relationship between i and ii.

Solution: i. P Ç
Q means the elements common to both sets

P Ç Q = {5}

(P
Ç Q)

^{ /}means the elements in the universal sets but not in the set PÇQ Þ (P ÇQ)^{ /}= {2, 3, 7}.
ii. P

^{/}= {3, 7} and Q^{/}= {2, 3}
P

^{/ }U Q^{/}= {2, 3, 7}
iii. The relationship between i and ii is they are equal sets.

Example
1.3 If μ = {3, 6, 9, 12, 15, 18, 21,
24, 27}, N = {3, 6, 9, 12, 15} and

M = {3, 9, 27}. Find (N È M)

^{/}where N, M Î μ.
Solution N È M = {3, 6, 9, 12, 15, 27}

(N È M)

^{/}= {18, 21, 24}
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