September 28 – Set Theory
Universal Set
A universal set (usually denoted by U) is a set which has elements of all the related sets, without any repetition of elements. Say if A and B are two sets, such as A = {1,2,3} and B = {1,a,b,c}, then the universal set associated with these two sets is given by U = {1,2,3,a,b,c}.
Some More Notation- Element or Not an Element
An element in a set is represented by the symbol 
If something is not an element in a set we use 
.
A, but 5 AEquality = Equal Sets
Two sets are equal if they have precisely the same members. The equal sign (=) is used to show equality, so we write:
A = B
Example: Are these sets equal?
- A is {1, 2, 3}
- B is {3, 1, 2}
Yes, they are equal!
They both contain exactly the members 1, 2 and 3.
It doesn’t matter where each member appears, so long as it is there.
What are Equivalent Sets? ↔
In general, we can say, two sets are equivalent ↔ to each other if the number of elements in both the sets is equal. It is not necessary that they have same elements, or they are a subset of each other.
Example:
If A = {1,2,3,4} and B = {♠♥♣♦} we can say set A is equivalent ↔ to set B because they have the same number of elements. That is because the cardinal value of both sets is 4. They symbol for equivalent sets is ↔
Equal sets on the other hand must have the same cardinal value and the elements must be the same even if not in the same order.
Subsets ⊂
When we define a set, if we take elements of that set, we can form what is called a subset.
Example: the set {1, 2, 3, 4, 5}
A subset of this is {1, 2, 3}. Another subset is {3, 4} or even another is {1}, etc.
But {1, 6} is not a subset, since it has an element (6) which is not in the parent set.
In general:
A is a subset of (⊂ ) B if and only if every element of A is in B.
So let’s use this definition in some examples.
Example: Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?
YES! A 
B since all the elements of A are also in B
If A is not a subset of B we write it as A 
B.
Empty (or Null) Set
A null set is a set with no elements.
This is known as the Empty Set (or Null Set).There aren’t any elements in it. Not one. Zero.
It is represented by the symbol 
Or by { } (a set with no elements)
The empty set is also a subset of every set.
Order
In sets it does not matter what order the elements are in.
Union Of Sets
The union of two sets A and B is the set of elements, which are in A and B . It is denoted by A ∪ B and is read ‘A union B’.
Example:
Given U = {1, 2, 3, 4, 5, 6, 7, 8, 10}
X = {1, 2, 6, 7} and Y = {1, 3, 4, 5, 8}
Find X ∪ Y and draw a Venn diagram to illustrate X ∪ Y.
Solution:
X ∪ Y = {1, 2, 3, 4, 5, 6, 7, 8} ← 1 is written only once.
Sets: Union And Intersection
∪ is the union symbol and can be read as “or”. The union of two sets are all the elements form both sets.
∩ is the intersection symbol and can be read as “and”. The intersection of two sets are those elements that belong to both sets.
The intersection of two sets are those elements that belong to both sets.
The union of two sets are all the elements from both sets.