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Math SETS – intersection - union - complement - Venn - diagrams

by Talia












































I would like for the following math questions to be answered.

1. What does N denote? (1mk)
2. Represent N in tabular form. (2mks)
3. What does Q denote? (1mk)
4. Represent Q in tabular form. (2mks)
5. What does I denote? (1mk)
6. Represent I in tabular form. (2mks)
7. What is another symbol that can be used instead of I? (1mk)
8. Describe what the following mean (Express in words) and represent in tabular form:
a) A= {x: x ε N, 3b)B= {x:x ε N, 2c) C= {x: x ε I, x is divisible by 4} (5mks)
d) D={x: x ε N, x > 50} (5mks)

9. Express the following in tabular form:
a) The set E is the set of all even numbers. (3mks)
b) The set F consists of the numbers 4, 5, 6 and 7. (3mks)
c) The set G consists of the numbers ½, ⅓ ¼, and ⅕. (3mks)
d) The set H consists of all girls in the college mathematic class on Wednesday September 2, 2009. (3mks)
e) The set I consists of all boys in the college mathematics class on Wednesday September 2, 2009. (3mks)
f) The set J consists of 2 girls and 1 boy in one college Mathematics class on Wednesday September 2, 2009. (3mks)
10. Are the following sets equal? Indicate your answer using the proper notation.
a) K= {1, 2, 3} and L= {3, 2, 1} (2mks)
b) M= {2, 5, 6} and N= (2, 5, 6, 6, 3} (2mks)
c) O= {John, Helen, Mary} and = {Mary, John, Helen} (2mks)
d) Q= {4, 1} and R= {3, 2} (2mks)
11. a) When are two sets equal? (2mks)
b) Give an example? (2mks)
12. a) What is a subset of a set? (2mks)
b) Give an example? (2mks)
13. a) What is a proper subset? (2mks)
b) Give an example? (2mks)
14. a) What is an improper subset? (2mks)
b) Give an example? (2mks)
15. a) What is a complementary set? (2mks)
b) Give an example? (2mks)
16. True or False (Indicate by writing True or False)
a) Ǿ is a subset of every set S. (1mk)
b) Ǿ is a proper subset of every set S ≠ Ǿ. (1mk)
17. a) What is a universal set? (2mks)
b) What is the complement of the universal set? (3mks)
18.a) What is the intersection of a set? (2mks)
b) Give the denotation. (1mk)
19. a)What is the union of two sets A and B? (2mks)
b) Give the denotation. (1mk)
c) What is the difference between sets A and B (A-B)? (2mks)
20. Represent the following in a Venn diagram by shading the area.
A. Given two sets A and B which have AΠB≠Ǿ
i) A (1mk) ii) B (1mk) iii) Ăс (2mks) iv) Bc (2mks) v) AΠB (2mks) vi) (AΠB)c (2mks)
vii) AUB (2mks) viii) (AUB) c (3mks) ix)AΠBc (4mks) x)BΠAc (4mks)

B. Given two disjointed sets A and B
xi) AUB (3mks) xii) Ac (2mks) xiii)Bc (2mks) xiv) (AUB)c (3mks)

C. Given two sets A and B
xv) ANB=Ǿ (2mks)












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Jan 29, 2011
Math SETS – intersection-union-complementary-Venn-diagrams
by: Staff



PART V

20. Represent the following in a Venn diagram by shading the area.

A. Given two sets A and B which have AΠB≠Ǿ


i) A (1mk)

set A

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-A.png



ii) B (1mk)

set B

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-B.png



iii) Ăс (2mks)

the absolute complement of A in the Universal set U
U ∖ A

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-A-absolute-complement.png



iv) Bc (2mks)
the absolute complement of B in the Universal set U
U ∖ B

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-B-absolute-complement.png


v) AΠB (2mks)
the intersection of sets A & B, A∩B

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-intersection-A-B.png


vi) (AΠB)c (2mks)
the absolute complement of the intersection of set A and set B

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-absolute-complement-intersection-A-B.png


vii) AUB (2mks)
the union of sets A & B, A∪B

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-union-A-B.png


viii) (AUB) c (3mks)
the absolute complement of the union of set A and set B

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-absolute-complement-union-A-B.png


ix)AΠBc (4mks)
the intersection of A and the absolute complement of B

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-intersection-A-absolute-complement-B.png


x)BΠAc (4mks)
the intersection of B and the absolute complement of A

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-intersection-B-absolute-complement-A.png


B. Given two disjointed sets A and B, A∧B

xi) AUB (3mks) – disjointed sets

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-disjoint-union.png






xii) Ac (2mks) – disjointed sets
the absolute complement set A

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-disjoint-complement-A.png


xiii)Bc (2mks) – disjointed sets
the absolute complement set B

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-disjoint-complement-B.png

xiv) (AUB)c (3mks) – disjointed sets
the absolute complement the union of set A and set B

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-disjoint-union-complement-A-B.png



C. Given two sets A and B


xv) ANB=Ǿ (2mks) – disjointed sets
N stands for ∩, intersection

Venn Diagram (click link to view):

http://www.solving-math-problems.com/images/Venn-set-disjoint-intersection-A-B.png



Thanks for writing.


Staff
www.solving-math-problems.com



Jan 29, 2011
Math SETS – intersection-union-complement-Venn-diagrams
by: Staff


PART IV


12. a) What is a subset of a set? (2mks)


“Subset” ⊆: Every element (without exception) contained in a subset must also be contained in the original set. . . . AND, the subset CAN BE EQUAL to the original set.

If the two sets are equal, each set is also called in “improper subset” of the other.



b) Give an example? (2mks)

For example, if set A = {1,11,12,53) and set C = {1,11,12,53), then set C is a “subset” of set A.

C ⊆A, C is a “subset” of A , even though it is contains exactly the same elements present in set A

C is also called in “improper subset” of A because the two sets are equal



13. a) What is a proper subset? (2mks)

“Proper Subset” ⊂: Every element contained in a subset must also be contained in the other set. . . . AND, the subset CANNOT BE EQUAL to the original set.



b) Give an example? (2mks)

For example, if set A = {1,44,60,150) and set B = {44,150), then set B is a “proper subset” of set A.

B ⊂A, B is a “proper subset” of A


14. a) What is an improper subset? (2mks)

See question 12a

b) Give an example? (2mks)

See question 12b

15. a) What is a complementary set? (2mks)

A complementary set is a set of things not in the original set.

A relative complementary set

a complement of a set refers to all elements outside the set.

Given sets A & B, the relative complement of A with respect to a set B, is the set of elements in B but not in A.

The absolute complement of A is the set of all elements in the universal set U, but not in set A.


b) Give an example? (2mks)

A = {a,b,c,d,e,f}
B = {a,b,c,d,e,f,g,h}

The complementary set of A with respect to B is: {g,h}


16. True or False (Indicate by writing True or False)


a) Ǿ is a subset of every set S. (1mk)

TRUE

b) Ǿ is a proper subset of every set S ≠ Ǿ. (1mk)

TRUE

17. a) What is a universal set? (2mks)

A set which contains all possible elements is called a universal set.

b) What is the complement of the universal set? (3mks)

An empty set: {}

18.a) What is the intersection of a set? (2mks)

The intersection of two sets includes only those elements found in both sets.

b) Give the denotation. (1mk)

A ∩ B

For example:

A = {cat, dog, hamster, fish, tree}

B = {fish, tree}

A ∩ B = {fish, tree}


19. a)What is the union of two sets A and B? (2mks)

The union of two sets includes all the elements found in A, AND all the elements found in the B.


b) Give the denotation. (1mk)
A ∪ B
For example:

A = {cat, dog, hamster, fish, tree}

B = {fish, tree, bird, snail}

A ∪B = {cat, dog, hamster, fish, tree, bird, snail }


c) What is the difference between sets A and B (A-B)? (2mks)

This is the complement of set B in set A.

A \ B

For example:

A = {cat, dog, hamster, fish, tree}

B = {fish, tree}

A-B = {cat, dog, hamster}


Jan 29, 2011
Math SETS – intersection-union-complement-Venn-diagrams
by: Staff




PART III


5. What does I denote? (1mk)

I: the imaginary part of a complex number: all numbers of the form n√(-1), or n*sqrt(-1)
(“I” does not stand for the set of all integers. The set of all integers is represented by “Z”)

An imaginary number is written as ni. The i stands for the sqrt(-1) .


6. Represent I in tabular form. (2mks)

There are too many imaginary numbers [of the form n√(-1)] to be expressed in tabular form.

7. What is another symbol that can be used instead of I? (1mk)

The Greek letter “Iota” ι is generally used to symbolize sqrt(-1), √(-1)
(Iota: alt+953; or type 03B9 on your keyboard immediately followed by alt+x)

The OTHER SYMBOL that is often used for √(-1) is j.


8. Describe what the following mean (Express in words) and represent in tabular form:

a) A= {x: x ε N, 3<12} (5mks)
I think you actually mean: A= {x: x ∈ N, x<12}
Set A consists of all x, where x is an element of set N (natural numbers) AND x is less than 12.

A= {1,2,3,4,5,6,7,8,9,10,11}



b)B= {x:x ε N, 2<4} (5mks)
I think you actually mean: A= {x: x ∈ N, x<4}

Set B consists of all x, where x is an element of set N (natural numbers) AND x is less than 4.

B= {1,2,3}


c) C= {x: x ε I, x is divisible by 4} (5mks)
I think you actually mean: A= {x: x ∈ I, x is divisible by 4}
(the set of integers is represented by Z, not I)

Set C consists of all x, where x is an element of set I (imaginary numbers) AND x can be divided by 4.

C= {4i,8i,12i, … }


d) D={x: x ε N, x > 50} (5mks)

Set D consists of all x, where x is an element of N (natural numbers) AND x is greater than 50.

D= {51,52,53,54,55, … }


9. Express the following in tabular form:
a) The set E is the set of all even numbers. (3mks)

E = {0,2,4,6,8, … }
(note: 0 is an even number because it is divisible by 2)



b) The set F consists of the numbers 4, 5, 6 and 7. (3mks)

F = {4,5,6,7}

c) The set G consists of the numbers ½, ⅓ ¼, and ⅕. (3mks)

G = {½, ⅓ ¼, ⅕}


d) The set H consists of all girls in the college mathematic class on Wednesday September 2, 2009. (3mks)

H = {girl}



e) The set I consists of all boys in the college mathematics class on Wednesday September 2, 2009. (3mks)

I = {boy}


f) The set J consists of 2 girls and 1 boy in one college Mathematics class on Wednesday September 2, 2009. (3mks)

J = {girl, boy}

10. Are the following sets equal? Indicate your answer using the proper notation.

a) K= {1, 2, 3} and L= {3, 2, 1} (2mks) - YES
{ 1,2,3} = { 3,2,1}
K = L

b) M= {2, 5, 6} and N= (2, 5, 6, 6, 3} (2mks) - NO
{ 2,5,6} ⊂ { 2,5,6,3}
M ⊂ N


c) O= {John, Helen, Mary} and P = {Mary, John, Helen} (2mks) - YES
{ John, Helen, Mary} = {Mary, John, Helen}
O = P


d) Q= {4, 1} and R= {3, 2} (2mks) - NO
{ 4,1} ≠ { 3,2}
Q ≠ R

11. a) When are two sets equal? (2mks)

Given two sets A and B.

If A is a subset of B, and B is also a subset of A, then the two sets are equal.

If A ⊆ B and B ⊆ A, then A = B


b) Give an example? (2mks)

A = {1,2,3} and B = {1,2,3}

A = B


Jan 29, 2011
Math SETS – intersection-union-complement-Venn-diagrams
by: Staff




PART II


The answer:
.

1. What does N denote? (1mk)
N: the set of natural numbers {1, 2, 3, 4, . . .}

The set of natural numbers (N) is a subset of integers (Z)

N ⊂ Z

2. Represent N in tabular form. (2mks)

N= {1,2,3,4,5, … }


3. What does Q denote? (1mk)
Q: the set of all rational numbers: {x/y : x ∈ Z, y ∈ Z }

The set of rational numbers (Q) is a subset of real numbers (R)

Q ⊂ R

4. Represent Q in tabular form. (2mks)

There are too many rational numbers to be expressed in tabular form. Select any two rational numbers – there will be another rational number between the two selected.

Any number that can be expressed as a ratio is a rational number.

Q = {a/b : a ∈ Z, b ∈ Z }



Jan 29, 2011
Math SETS – intersection-union-complement-Venn-diagrams
by: Staff


PART I

The question:
by Talia

I would like for the following math questions to be answered.

1. What does N denote? (1mk)
2. Represent N in tabular form. (2mks)
3. What does Q denote? (1mk)
4. Represent Q in tabular form. (2mks)
5. What does I denote? (1mk)
6. Represent I in tabular form. (2mks)
7. What is another symbol that can be used instead of I? (1mk)
8. Describe what the following mean (Express in words) and represent in tabular form:
a) A= {x: x ε N, 3<12} (5mks)
b)B= {x:x ε N, 2<4} (5mks)
c) C= {x: x ε I, x is divisible by 4} (5mks)
d) D={x: x ε N, x > 50} (5mks)

9. Express the following in tabular form:
a) The set E is the set of all even numbers. (3mks)
b) The set F consists of the numbers 4, 5, 6 and 7. (3mks)
c) The set G consists of the numbers ½, ⅓ ¼, and ⅕. (3mks)
d) The set H consists of all girls in the college mathematic class on Wednesday September 2, 2009. (3mks)
e) The set I consists of all boys in the college mathematics class on Wednesday September 2, 2009. (3mks)
f) The set J consists of 2 girls and 1 boy in one college Mathematics class on Wednesday September 2, 2009. (3mks)
10. Are the following sets equal? Indicate your answer using the proper notation.
a) K= {1, 2, 3} and L= {3, 2, 1} (2mks)
b) M= {2, 5, 6} and N= (2, 5, 6, 6, 3} (2mks)
c) O= {John, Helen, Mary} and = {Mary, John, Helen} (2mks)
d) Q= {4, 1} and R= {3, 2} (2mks)

11. a) When are two sets equal? (2mks)
b) Give an example? (2mks)
12. a) What is a subset of a set? (2mks)
b) Give an example? (2mks)
13. a) What is a proper subset? (2mks)
b) Give an example? (2mks)
14. a) What is an improper subset? (2mks)
b) Give an example? (2mks)
15. a) What is a complementary set? (2mks)
b) Give an example? (2mks)
16. True or False (Indicate by writing True or False)
a) Ǿ is a subset of every set S. (1mk)
b) Ǿ is a proper subset of every set S ≠ Ǿ. (1mk)
17. a) What is a universal set? (2mks)
b) What is the complement of the universal set? (3mks)
18.a) What is the intersection of a set? (2mks)
b) Give the denotation. (1mk)
19. a)What is the union of two sets A and B? (2mks)
b) Give the denotation. (1mk)
c) What is the difference between sets A and B (A-B)? (2mks)
20. Represent the following in a Venn diagram by shading the area.
A. Given two sets A and B which have AΠB≠Ǿ
i) A (1mk) ii) B (1mk) iii) Ăс (2mks) iv) Bc (2mks) v) AΠB (2mks) vi) (AΠB)c (2mks)
vii) AUB (2mks) viii) (AUB) c (3mks) ix)AΠBc (4mks) x)BΠAc (4mks)

B. Given two disjointed sets A and B
xi) AUB (3mks) xii) Ac (2mks) xiii)Bc (2mks) xiv) (AUB)c (3mks)

C. Given two sets A and B
xv) ANB=Ǿ (2mks)

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