Transcript May 2001: Paper 2 #1

May 2001: Paper 2 #1
The sets A, B, and C are subsets of U. They are defined as follows:
U = {positive integers less than 16}
A = {prime numbers}
B = {factors of 36}
C = {multiples of 4}
(a) List the elements (if any) of the following:
Before we do any of these, let's make sure we understand the sets. A, B, and C are
subsets of U.
This makes U are universal set. In other words, our "universe" consists only of positive
integers less than 16. (NOTE: 0 is not positive or negative, and 16 is not included!)
So our universe is all integers between 1 and 15.
A is the set of all prime numbers between 1 and 15 (NOTE: 1 is NOT prime!)
B is the set of all the factors of 36 that are between 1 and 15.
Lastly, C is the set of multiples of 4 that are between 1 and 15.
May 2001: Paper 2 #1
The sets A, B, and C are subsets of U. They are defined as follows:
U = {positive integers less than 16}
A = {prime numbers}
B = {factors of 36}
C = {multiples of 4}
(a) List the elements (if any) of the following:
So,
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
A = {2, 3, 5, 7, 11, 13}
B = {1, 2, 3, 4, 6, 9, 12}
C = {4, 8, 12}
And, A Ç B Ç C is the set containing elements that are in ALL 3 sets A, B,
and C. Since there does not exist any elements in ALL 3 sets we say
AÇ B ÇC = Æ
May 2001: Paper 2 #1
(b)
(i) Draw a Venn diagram showing the relationship between the sets U, A, B, and C.
Since we just decided that the intersection of all 3 sets A, B, and C was empty we
do not need to draw the Venn diagram where all three sets intersect. However,
upon further inspection, we should notice that A and B share elements and B and
C share elements. The reason that the intersection of all three is empty is because
A and C do no share any elements. Thus, the Venn Diagram should look like
BE SURE TO LABEL ALL FOUR SETS FOR FULL CREDIT!!!!
U
A
B
C
May 2001: Paper 2 #1
(b)
(ii) Write the elements of sets U, A, B, and C in the appropriate places on the Venn
diagram. Let's see the Venn Diagram and the List of Elements.
U
B
A
C
2
5
14
13
10
15
7
3
11
6
4
9
12
8
1
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
A = {2, 3, 5, 7, 11, 13}
B = {1, 2, 3, 4, 6, 9, 12}
C = {4, 8, 12}
May 2001: Paper 2 #1
(c) From the Venn Diagram, list the elements of each of the following
(i) A Ç (B È C ) =Means
{2 , 3}all elements in B or C together (union).
U
B
A
C
2
5
14
13
10
15
7
3
11
6
4
9
12
8
1
Now, intersect what is shaded with set A.
May 2001: Paper 2 #1
(c) From the Venn Diagram, list the elements of each of the following
Means
"complement";
all 13,
elements
"NOT" in ( A Ç B )
(ii) ( A Ç B )' = {1, 4,
5, 6, 7,
8, 9, 10, 11, 12,
14, 15}
U
B
A
C
2
5
14
13
10
15
7
3
11
6
4
9
12
1
8
May 2001: Paper 2 #1
(c) From the Venn Diagram, list the elements of each of the following
(iii) (A Ç B )' Ç C = {4,
From
the previous problem…
8, 12}
U
B
A
C
2
5
14
13
10
15
7
3
11
6
4
9
12
8
1
Now, intersect what is shaded with set C
May 2001: Paper 2 #1
(d) Find the probability that a number chosen at random from the
universal set U will be
(i) a prime number**
U
B
A
C
2
5
14
13
10
15
P (**) =
7
3
11
6
4
9
12
8
1
Favorable Outcomes
Prime Numbers
6
2
=
=
=
Possible Outcomes
Numbers in Universal Set 15 5
May 2001: Paper 2 #1
(d) Find the probability that a number chosen at random from the
universal set U will be
(ii) a prime number, but not a factor of 36**
U
B
A
C
2
5
14
13
10
15
P ** =
7
3
11
6
4
9
12
8
1
Favorable Outcomes
Prime Numbers NOT a factor of 36
4
=
=
Possible Outcomes
Numbers in Universal Set
15
May 2001: Paper 2 #1
(d) Find the probability that a number chosen at random from the
universal set U will be
(iii) a prime number, but not a factor of 36**
U
B
A
C
2
5
14
13
10
15
7
3
11
6
4
9
12
8
1
Favorable Outcomes
Possible Outcomes
Factor of 36 OR Multiple of 4, BUT NOT a Prime
6 2
=
=

Numbers in Universal Set
15 5
P ** =
May 2001: Paper 2 #1
(d) Find the probability that a number chosen at random from the
universal set U will be
(iv) a prime number given that it is a factor of 36
U
B
A
C
2
5
14
13
10
15
P ** =
7
3
11
6
4
9
12
8
1
Favorable Outcomes
Prime AND a Factor of 36 2
=
=
Possible Outcomes
Factors of 36
7