1. Session – 2 SETS & Operations of SETS

2. Subset and Proper Subset
Subset : Let A and B be two sets. Then A is said to be a subset of B if every element of A is an element of B.
It is represented by “ “
Proper Subset : A is said to be a proper subset of B if A is a subset of B and there is at least one element of B which is not in A.
It is represented by “ “

3. Two sets A and B are equal iff
Property :
Two sets A and B are equal iff
NULL SET
Singleton Set
We write A = B.
A set containing no elements is called the empty set or null set, denoted by
A set containing only one element is called Singleton set.

4. Exercise List all the subsets of { Java, C++, VB }.
{ Java } , { C++ }, { VB }, { Java, C++ },{ Java, VB }, { C++, VB },{ Java, C++, VB }, .
List all the subsets of the set .
 or {}

5. Exercise b. { 3 } Í S c. 2 Î S d. { 2 } Ï S e. { Æ } Î S f. Æ Î S
Let S = { { 3 }, { 2 }, { 3, 2, 1 }, { Æ } }
Which of the following are true. Be careful again, set of sets!
a. { 3 } Î S
b. { 3 } Í S
c. 2 Î S
d. { 2 } Ï S
e. { Æ } Î S
f. Æ Î S

6. Answers
a True
b False
c False
d False
e True
f False

7. Exercises
1. Which of the following statements are true?
a. 5  { 3,1,2 }
b. 1  { 3,1,2 }
c. 5  { 3,1,2 }
d. { 3 }  { 3,1,2}
e. { 3,1,2 }  { 3,1,2 }
f. { 2,1 }  { 3,1,2 }
g. {}  { 3,1,2 }
h. { 3 }  { 3,1,2 }
i. { 3,1,2 }  { 3,1,2}

8. a False b True c True d False e True f True g True h True i False
Solutions
a False
b True
c True
d False
e True
f True
g True
h True
i False

9. Exercises
1. Which of the following statements are true?
j. { 2,1 }  { 3,1,2 }
k.   { 3,1,2 }
l. { 3,1,2 } = { 3,1,2 }
m. { a,b,a,c } = { a,b,c }
n. { a,b,a,c } ≠ { c,a,b,b }

10. Solutions
j True
k True
l True
m True
n False

11. Exercises 2. Let A = { 5, 2, 4, a, b, d }. Which of the following are true? a. d  A b. 3  A c. c  A d.   A e.   A f. A  A

12. Solutions
a True
b False
c True
d False
e True
f False

13. Exercises
Which of the following statements are true and why or why not?
1. { b, a }  { c, a, b }
2. { b, d }  { c, a, b }
3. { b, a }  { c, a, b }
4. {} = { {} }
5. {}  {}
6. {}  {}
7. {}  {this set }
8. {}  { {} }
9. { {} }  {}
10. { a,b } = ( a,b )
11. { { a,b } } = { (a,b) }

14. Answers :
1. is true as the first set contains elements a and b and so does the second.
2. is false as the first set contains an element d which the second set does not.
3 is true as the first set contains elements a and b and so does the second but the second also contains another element and so is larger.
4. is false as the empty set is not the same as the set that contains the empty set.
5. is true as the empty set is a subset of every set (even itself).
6. is false as no set can be a proper subset of itself.
7. is true for same reason as 5.
is true for same reason as 5 and 7.
9. is false as the first set contains the empty set and will be a bigger set than the empty set.
10. is false as (a,b) is not the set {a,b}, it is in fact a pair (more about pairs later).
11. is false as the first set contains the set {a,b} and the second set contains the pair (a,b).

15. Operations on Sets
Union
Intersection
Symmetric difference
Complement

16. Union

17. Intersection

18. Symmetrical Difference

19. Complement of a Set

20. Basic properties of Union and Intersection of two sets.

21. Distributive Laws

22. Demorgan’s Laws

23. Disjoint

24. Venn Diagrams

25. Power Set