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Basic Set Theory

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Mathematics

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Study Guide

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Basic Set Theory
Sets are written in curly brackets with a comma between each member or element inside. To make it
easy to grasp the concepts, we will work in a familiar setting – the school. Suppose there are 11
students in the whole school. Let us call them a, b, c, d, …k. In set notation we write this as :
U= {a,b,c,d,e,f,g,h,i,j,k}
The letter capital U usually stands for the universal set because it is the universe or total of all the
students in our school - there are no more. Also, the order of writing the elements does not matter.
There are 11 elements or members in this universal set (we would call them students if we were
outside the mathematics class). A subset of U (or simply a set in U) is just a set with some of the
members we use (capital) letters for them. Three sets are shown below.
A={a,b,c,d,e}
B= {c,d,e,f}
C= {h,i,j}
Now we will define three operations on these sets. Operations are just a way of combining the sets.
Consider AB (read as A intersect B)
The intersection of two sets is simply the set containing all the elements common to both.
Hence AB = {c,d,e}
We could also write {a,b,c,d,e}{c,d,e,f} = {c,d,e}
Now suppose set A represents the members of the football team and B represents the rugby team. The
headmaster wants to see all members of the football team and all members of the rugby team in his
office immediately. Who should go? Obviously a,b,c,d,e,f should all go.
We represent this situation as A B (read as A union B)
The Union of two sets is the set containing all the elements present in the two sets. Notice we don't
repeat members: c,d,e play both sports, but they can only be present once!
Hence AB = {a,b,c,d,e,f} or we could write{a,b,c,d,e}{c,d,e,f} = {a,b,c,d,e,f}
Notice we don't repeat members: c,d,e play both sports, but they can only be present once!
Now for the third operation, called complement (or prime) of a set.
The complement of set A is written A'.
Suppose A ={a,b,c,d,e} Think of the elements who happen to be present in school on a particular day.
Now who were absent from school? To find out, we consult the school roll (the universal set U):
U = {a,b,c,d,e,f,g,h,i,j,k}
A={a,b,c,d,e} (These are present)
A'= {f,g,h,i,j,k} (These are absent)
Similarly
U = {a,b,c,d,e,f,g,h,i,j,k}
B = { c,d,e,f}

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B' = {a,b,g,h,i,j,k}
Similarly U = {a,b,c,d,e,f,g,h,i,j,k}
C = { h,i,j}
C' = {a,b,c,d,e,f,g,k}
The formal definition of the complement of a set is as follows:
Let A be a subset of the Universal set U
Then A' is the set containing all the elements in U which are not in A.
You have now met all three basic operations. Now it's time for a little exercise;
Exercise If U = {1,2,3,4,5,6,7,8,9}
C = {1,2,3,4,5,6}
D= {4,5,6,7,8}
and G= {7,8,9}
Find
a) C∩D and D∩C
b) CD and DC
c) C C and C ∩ C
d) C G and CG
e) C'
f) G'
(Solution on the next page)

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Solution
a) C∩D ={1,2,3,4,5,6}∩ {4,5,6,7,8}
={4,5,6,}
D∩C ={4,5,6,7,8}∩ 1,2,3,4,5,6}
={4,5,6,}
Obviously the answer is the same because we are finding what elements are common to both sets, so
the order of writing the two sets does nor matter.
In general, for any sets A and B, A ∩ B = B ∩ A
b) Now the second part
CD ={1,2,3,4,5,6}{4,5,6,7,8}
={1,2,3,4,5,6,7,8}
and DC = {4,5,6,7,8} {1,2,3,4,5,6}
={4,5,6,7,8, 1,2,3,4,5,6} or {1,2,3,4,5,6,7,8}
Obviously the answer is the same because we are combining the two sets by putting all the elements
together – the Set order does not matter
In general, for any sets A and B, A B = B A
C) C∩C ={1,2,3,4,5,6}∩ {1,2,3,4,5,6}
={1,2,3,4,5,6}
= C
and CC ={1,2,3,4,5,6}{1,2,3,4,5,6}
= {1,2,3,4,5,6}
= C
In general, for any set A
A A =A
d) C G and CG
C G ={1,2,3,4,5,6}∩ {7,8,9}
There are no elements that the two sets have in common !
We write {1,2,3,4,5,6}∩ {7,8,9} = { } leaving the elements blank
or use a special symbol Ø to stand for the empty set – the set with no members.
Note the empty set is not the same as {0} because this set is not empty. It has an element (zero) inside
it, just like {1} or {a} .

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Basic Set TheorySets are written in curly brackets with a comma between each member or element inside. To make it easy to grasp the concepts, we will work in a familiar setting - the school. Suppose there are 11 students in the whole school. Let us call them a, b, c, d, ?k. In set notation we write this as : U= {a,b,c,d,e,f,g,h,i,j,k}The letter capital U usually stands for the universal set because it is the universe or total of all the students in our school - there are no more. Also, the order of writing the elements does not matter.There are 11 elements or members in this universal set (we would call them students if we were outside the mathematics class). A subset of U (or simply a set in U) is just a set with some of the members we use (capital) letters for them. Three sets are shown below.A={a,b,c,d,e}B= {c,d,e,f}C= {h,i,j}Now we will define three operations on these sets. Operations are just a way of combining the sets.Consider A?B (read as A intersect B)The intersection of two sets is simply the set containing all the elements common to both.Hence A?B = {c,d,e}We could also write {a,b,c,d,e}?{c,d,e,f} = {c,d,e}Now suppose set A represents the members of the football team and B represents the rugby team. The headmaster wants to see all members of the football team and all members of the rugby team in his office immediately. Who should go? Obviously a,b,c,d,e,f should all go.We represent this situation as A? B (read as A union B)The Union of two se ...
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