Download Number Sets Review #2 File

Set Builder
Notation
If you recall, a set is a collection of objects which we write using brackets
and name using a capital letter.
Remember also that:
 means is an element of.
 means is NOT an element of.
 or { } means the empty set and contains no elements.
Also a set is a subset of another if every element in one set is also an
element in the other set.
To write this, we use the notation

Lastly, remember that the intersection of two sets, or groups is the
elements that both groups have in common.
The word AND is used to mean an intersection. You can also use the
symbol:
Finally, the union of two sets, or groups is the combination of the elements
in both groups. It doesn’t matter if they are in common or not.
The word OR is used to mean a union. You can also use the symbol:
We can count the number of elements in a set.
We use the notation n(A) to represent the number of elements in set A.
For example:
The set {1, 2, 3, 4, 5} has five elements in it, so we can write n(A) = 5.
We read this as the
number of elements
in A is 5.
A finite set has a finite number of elements in it (a countable quantity).
For example:
The set of all positive integers less than 100 is a finite set because we
can count the exact number.
n( A)  99
An infinite set has infinitely many elements in it (an uncountable quantity).
The set of all integers is an infinite set because we can never count the
exact number of integers total.
n( Z )  
Set Builder Notation is a way that we can describe sets using symbols.
It looks complicated but it is a way to make our work easier.
{}
means “the set of”
or :
means “such that”
,
means “also”
So, we get a statement that looks like:
A  {x x  Z ,  2  x  4}
which we translate to:
A is the set of all x such that x is an element of the integers, also x is
between -2 and 4, including -2 and 4.
Translate the following statements:
B  {x x  R, 0  x  1}
B is the set of all x such that x is an
element of the real numbers, also x is
between 0and 1, including 0 and 1.
A  {x : x  Z , x  0}
A is the set of all x such that x is an
element of the integers, also x is
greater than 0.
G  {x : x  Z ,1  x  2}
G is the set of all x such that x is an
element of the integers, also x is
between 1and 2, including 1and 2.
D  {x x  Q, x  1}
D is the set of all X such that x is an
element of the Rational Numbers,
also x is less than -1, including -1.
State if the following sets are finite or infinite:
B  {x x  R, 0  x  1}
A  {x : x  Z , x  0}
G  {x : x  Z ,1  x  2}
B is an infinite set since it includes all
real numbers between 0 and 1.
A is an infinite set since it includes all
integers greater than 0.
G is a finite set since there are only
two numbers that work. They are 1
and 2.
Given that:
A  {x : x  Z , 3  x  10}
write down:
a) the meaning of the set in words.
b) the elements that are found in set A.
c) n(A).
Answer:
a) A is the set of all x such that x is an integer,
also is between 3 and 10 including 10.
b) A = {4, 5, 6, 7, 8, 9, 10}
c) n(A) = 7
Given that:
B  {x : x  Z , 1  x  4}
write down:
a) the meaning of the set in words.
b) the elements that are found in set B.
c) n(B).
Answer:
a) B is the set of all x such that x is an integer,
also x is between -1 and 4.
b) B = {0, 1, 2, 3}
c) n(B) = 4
Given that:
A  {x : x  Q,1  x  3}
write down:
a) the meaning of the set in words.
b) the elements that are found in set A.
c) n(A).
Answer:
a) A is the set of all x such that x is an rational
number, also is between 1 and 3 including 3.
b) Impossible to list since this is an infinite set.
c) n(A) = infinite
Given that: B
 {x : x  N ,  3  x  3} write down:
a) the meaning of the set in words.
b) the elements that are found in set B.
c) n(B).
Answer:
a) B is the set of all x such that x is an natural
number, also x is between -3 and 3.
b) B = {0, 1, 2}
c) n(B) = 3
Given that:
A  {2, 5, 7}
and
B  {2, 3, 4, 5, 6, 7, 8} is A  B ?
Answer:
Yes A is a subset of B since every element of A is
also in B.
Given that: A  {x : x  Z ,  2 
is A  B ?
x  2} and B  {x x  Z ,  2  x  2}
Answer:
Yes A is a subset of B since every element of A is
also in B.
Universal Sets –
The main set for the given situation (subsets will be selected from this set).
The symbol U is used to represent a universal set.
Complementary Sets –
This is used to represent everything that is not in a particular set.
The complement of A, with symbol A’, is the set of all elements of U
which are not in A.
The relationships connecting A and A’ are:
A  A'  
A  A'  U
n( A ')  n( A)  n(U )
Here is an example:
Given that
down A’.
U  {1, 2, 3, 4, 5, 6, 7, 8} and A  {1, 3, 5, 7, 8} , write
Answer:
A’ = {2, 4, 6}
Given that
down A’.
U  {all integers} and A  {all even integers}, write
Answer:
A’ = {all odd integers}
Given that
U  {Z } and A  {x : x  Z , x  2} , write down A’.
Answer:
A '  {x : x  Z , x  1}
Given that U  {all
write down A’.
real numbers} and A  {all rational numbers} ,
Answer:
A’ = {all irrational numbers}
Given that
and
U  {x : x  Z ,  5  x  5} and A  {x : x  Z , 1  x  4}
B  {x x  Z ,  3  x  2} list the elements of the following sets:
a) A
b) B
Answer:
A  {1, 2,3, 4}
c) A’
B  {3, 2, 1, 0,1}
d) B’
A '  {5, 4, 3, 2, 1, 0,5}
e)
A B
B '  {5, 4, 2,3, 4,5}
A  B  {1}
f)
A B
A  B  {3, 2, 1, 0,1, 2,3, 4}
g)
A ' B
A ' B  {3, 2, 1, 0}
h)
A ' B '
A ' B '  {5, 4, 3, 2, 1, 0, 2,3, 4,5}
Given that
and
U  {x : x  Z ,  3  x  4}
and
A  {x : x  Z , 1  x  2}
B  {x x  Z ,  3  x  0} list the elements of the following sets:
a) A
b) B
Answer:
A  {1, 2}
c) A’
B  {3, 2, 1, 0}
d) B’
A '  {3, 2, 1, 0,3, 4}
e)
A B
B '  {1, 2,3, 4}
A B  
f)
A B
A  B  {3, 2, 1, 0,1, 2}
g)
( A  B)  A '
( A  B)  A '  {3, 2, 1, 0,1, 2,3, 4}
h)
( A  B)  B '
( A ' B)  B '  {3, 4}
Given that U  {positive integers}
and A  {multiples of 4 that are less than 50}
and B  {multiples of 6 that are less than 50} find the following:
a) A
b) B
c)
A B
d)
A B
e) Verify that
Answer:
A  {4,8,12,16, 20, 24, 28,32,36, 40, 44, 48}
B  {6,12,18, 24,30,36, 42, 48}
A  B  {12, 24,36, 48}
A  B  {4, 6,8,12,16,18, 20, 24, 28,30,32,36, 40, 42, 44, 48}
n( A  B )  n( A)  n( B)  n( A  b)
so we get 16  12  8  4
n( A  B)  n( A)  n( B)  n( A  B)