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 2ⁿ.

For example a set of 2 elements will have 2² 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 = 2ⁿ, where n is the number of elements in the set i.e.

n = 3. So we have 2³ = 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}