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SIXTH GRADE / BASICS of SET THEORY / TEST 2.1.1

MATHEMATICS IS EASY. PROVIDED THAT ONE WORKS ENOUGH.
ENOUGH IS SKILL DEPENDENT, HAVE TO FIND "ENOUGH" FOR YOUR SIZE.

SOME OF OUR QUESTIONS ARE REALLY HARD, FOR THOSE THAT ARE SKILLED IN MATH.
SKIP THEM IN CASE NOT FOR YOU.

IF YOU DESIRE SO, WE CAN DISCUSS HARD/INTERESTING QUESTIONS IN OUR FORUM.

  SET THEORY
 ♦ 
IS THE FIRST ABSTRACT SUBJECT in MATH.  SOME TERMS of IMPORTANCE are ELEMENT, INTERSECTION, UNION, EMPTY SET, UNIVERSAL SET, COMPLEMENT.
 ♦ IDEMPOTENT, COMMUTATIVE, ASSOCIATIVE, DISTRIBUTIVE, IDENTITY, COMPLEMENT, De MORGAN'S LAWS and CONSISTENCY PRINCIPLE are INPORTANT PROPERTIES.

THEY FORM THE FOUNDATION of THE THORY of PROBABILITY which is THE TRUE LOGIC of THE WORLD.

    Ref.-1         Ref.-2
1. Which of the followings define a set ?

1- Natural numbers.
2- Binary sums of all natural numbers.
3- Clever boys in our school.
4- Sum of two naural numbers.


a) only 1
b) 1 and 2
c) only 2
d) 2 and 3
Definitions of the sets should permit its elements to be identified. Binary sums of all natural numbers are again natural numbers but sum of two natural numbers is vague .
2. What is the number of elements of the set {a, a, b, c, d} ?

a) 5
b) 3
c) 4
d) 2
This set has elements a, b, c ve d ; even if the element a is written twice, it is only one element .
3. What are the elements of the big set within the red rectangle in the figure ?

!


a) 6
b) 3
c) 3 and 6
d) 3 and {6}
A set may be an element of another set .
4. Which elemnets are in the union of the sets within the blue rectangle below in the figure ?

!


a) 4, 5, 7, 8, and 8
b) 5, 7, 8, and {8}
c) 4, 5, 7, 8, and {8}
d) 5, 7, 8, and 8
There are 4 sets in the region .
The elements of the sets in this region are 4, 5, 7, 8, and {8} olmaktadır.
5. Which choice gives the intersection of the sets within the red rectangle of figure of question-3 ?

a) rectangle
b) complement
c) universal set
d) empty set=∅
There are two sets in this region : A = {3, {6}} ve B = {6} . 6 is an element of B, whereas {6} is of A; no common element .
y
6. Which choice gives the number of elements of the set that contains 2 and the sets within the blue rectangle of figure of question-4 ?

In case of need for tips get the cursor on "y" or for help click at "ry1/2" .
RY1 RY2



a) 512
b) 256
c) 128
d) 454


The empty set and the set itself are among the subsets of any set.
A set having N elements has 2N subsets. This is explained in ry1 for a set with N=3 .
The question is hard; we give the elements: 2, {4}, 4, {5}, 5, {7,{8}}, 7, {8} and 8.
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7. Which of the following statements are true ?

1- There are only two main ways to specify a set: Listing all its members (list notation) and giving a Venn diagram.
2- Sets having equal cardinalities are identical .
3- Identical sets have equal cardinalities.
4- It is also possible to specify a set by stating a property of its elements (predicate notation) .

a) 1 and 3
b) 2 and 4
c) 3 and 4
d) 1 and 4
Sets having equal cardinalities have the same number of elements but not more.
8. Which set has the greatest number of elements ?

a) {Days beginning from August 1, 2000}
b) {Days of the week}
c) {letters of the word "Caroline"}
d) {The novels by Mark Twain}
The number of the novels by M. Twain is less than 15 .
9. A = {1, 2, 3} is given. How many of its subsets do have (only) one element ?

a) 4
b) 3
c) 2
d) 1
To see the subsets, have a look at ry1 of question-6 .
10. It is given that the area of the union of the rectangles, A(ABCD ∪ EFGH), is 66 cm2. How big is the area of the green region, the intersection of them ?

!


a) 16 cm.2
b) 14 cm.2
c) 6 cm.2
d) 9 cm.2
The areas of the rectangles are 40 cm.2 and 35 cm.2 ... .
11. What will be the number of elements of a set A if the the sum of the number of subsest and proper subsets is 15 ?

a) 4
b) 3
c) 2
d) 5
If A ⊆ B and A ≠ B we call A a proper subset of B and write A ⊂ B , i.e. the number of proper subsets = number of subsets - 1 .
y
12. Which choice is the green filled area if A = {Points of rectangle ABCD}, I = {Points of rectangle IJKL} ve E = {Points of rectangle EFGH} ?

!


In case of need for tips get the cursor on "y" or for help click at "ry1/2".

RY1 RY2



a) A ∩ I
b) (E ∖ A) ∪ I
c) (E ∖ A) ∩ I
d) A ∪ I
Another binary operation on arbitrary sets is the difference “A minus B”, written A – B or A\B, which ‘subtracts’ from A all elements which are in B.
13. Another binary operation on arbitrary sets is the difference “A minus B”, written A – B or A\B, which ‘subtracts’ from A all elements which are in B. Which choice specifies it by stating a property of its elements the right way ?

a) elements in A but not in B
b) A \ B = { x | x ∈ A ve x ∉ B}
c) elements in B but not in A
d) A \ B = { x | x ∈ B ve x ∉ A}
x ∈ A means x is in A , x ∉ A means x is not in A .
14. Which choice is wrong according to the Venn diagram ?

!


a) none
b) 4 ∈ A ∩ B
c) ∅ = A ∩ C
d) ∅ = B ∩ C


Not the boundaries of the Venn diagram but the elements specify the sets .
15. Which choice is true for the set A = {SOUTH CAROLINA} ?

a) s(A) = 1
b) SOUTH ∈ A
c) CAROLINA ⊂ A
d) ∅ = A
Different elements of a set are separated by commas .
16. The sets A = {1, 2, 3}, B = {3, 5, 7} and C = {7, 3, {5}} are given. Which is/are wrong ?

1- A and B have equal cardinalities.
2- A and C are identical.
3- B and C are identical.
4- 5 ∈ C

a) all
b) 1 and 3
c) 2, 3 and 4
d) 2 and 4
Not 5 but the set having 5 as the only element is in C .
17. The set A = {NORTH, C, A, R, O, L, I, N, A} is given. Which choice is a subset of A ?

a) {NORTH, O, L, I, N, A}
b) {NORTH, CAR}
c) {C, A, R, O, L, I, N, E}
d) C, A
Each element of a subset must be an element of the set .
y
18. What will be the number of 2-element subsets of one of the sets A or B, if the cardinalities are equal and s(A ∩ B) = 4 and s(A ∪ B) = 10 ?

In case of need for tips get the cursor on "y" or for help click at "ry1/2" .
RY1 RY2


a) 21
b) 18
c) 15
d) 12
s(A) + s(B) = 4 + 10 = 14 and equal cardinalities mean each set must have 7 elements .
Have please a look at ry1 .
19. Which choice is correct according to the figure below ?

!


a) {5} ∈ A
b) {15} ∈ A
c) 5 ⊂ A
d) 35 ∉ A
Subset of a set is again a set .
20. Which choice gives the number of elements of B and C, if we separate the set A according to the number of tornados being less or greater than 10, name the greaters as B and the less as C ?

!


a) 4 and 4
b) 4 and 6
c) 4 and 3
d) 5 and 5
14 and 4 are not in A .
21. Which choice(s) is/are correct according to the figure given in the previous question ?

1- {5,29} ⊄ B
2- A ∩ B = C
3- B ∩ C ≠ ∅
4- B ∪ C = A
5- 3 ∈ C

a) 1 and 3
b) 4 and 5
c) only 1
d) only 4
⊄ means not a subset of .
y
22. How many 3-membered subsets (subsets with 3 elements) should A have ?

!


In case of need for tips get the cursor on "y" or for help click at "ry1/2" .

RY1 RY2



a) 8
b) 16
c) 21
d) 35


The number is pretty big, have a look at ry1. There we listed them.
23. The first 8 rows of Pascal Triangle is given below. In which row are the numbers 21 and 35 that give the 2-membered and 3-membered subsets of a 7-element set ?

!


a) 2nd
b) 5th
c) the one that has a 6 in 2. column
d) 8th that has a 7 in 2. column
24. Which choice gives the set of prime numbers in the green colored region ?

!


a) {3,5,7,11,13,17,31,37,43}
b) {3,5,7,13,17,31,37,43}
c) {3,5,7,11,13,17,19,23,31,37,43}
d) {1,3,5,7,13,17,31,37,43}
1 is not prime .
25. How many proper subsets does the set of prime numbers in the white colored region of the figure of the previous question ?

a) 32
b) 16
c) 24
d) 8
25 = ?
This is the feedback!

THANK YOU FOR YOUR VISIT; HOPE WE COULD BE OF USE.
FOR MUTUAL BENEFIT YOU MAY USE OUR FORUM.
© 2012 , İsmail GERMAN. ALL RIGHTS RESERVED
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