Overall it looks good.
The parts highlighted with yellow are incorrect.
The parts highlighted with blue are correct but are to be moved elsewhere.
The sentences in red are my comments .
Sets
What is a Set ?
Set is a group of elements, having a property that characterizes those elements.
How to specify a Set ?
One way is to enumerate the elements completely ( all the elements belonging to set are explicitly given) and give a property that characterizes the elements of the Set.
Example: A = {1,2,3,4,5}
Other way is to enumerate the elements in a way that makes clear what the remaining elements are and to give a property that characterizes the elements of the Set.
Example: B = {x/x is a positive integer less than or equal to 5}
C = {1,2,3,4,5,6,……..} ( this is a n infinite set)
Add "Recursive definition of sets" with an example.
How to play around with Sets? What do you mean by this ?

x E B implies that x is an element of set B. Using this notation we can specify the set {1,2,3,4,5} by writing
Z = {x

A \ x <= 5} { x | x

A and x <= 5 } which is read as “ the set of elements ‘x’ in ‘A’such that ‘x’<5”.
Subset:
If A and B are two sets,
A is a subset of B, if every element of A is an element of B.
Example: A = {1,2,3}
B = {x/x is a positive integer}
Here we can think of some better way of specifying the set B just by replacing the
English like sentence “x is a positive integer less than 100” in a more mathematically simpler way. We can achieve it by assuming ‘N’ as all positive integers.
We can specify set B as,
B = {x / x E N} If you use N, then you might as well say A

N .
1,2,3 are all clearly one of the elements in set B as they satisfy the property that characterizes the elements of set B. (they all are positive integers)
A is a subset of B is represented as A

B.
If A is a subset of B and B is a subset of A then A

B. Also, A is subset and equal to B represented as A

B and B

A.
Compliment: Complement
If A is a set, then the compliment of A is the set A’ consisting of all elements contained in the universal set and not in A.
Universal Set:
A set U that contains all the elements we might ever consider. Put this before
Complement
Then the complement of A, denoted by A’, can be represented as
A’={x 
U | x

A}
Where

“is not an element of”.
E.g.: if U={1,2,3….}
A= {1,2,3} then A’={4,5,6….}
The operations that can be performed on sets are:

1.
Union: If A and B are two sets, then A union B is a set that contains all the elements only in A, only in B and in both A and B which can be represented as
A

B.
E.g.: A={1,2,3} B={3,4,5} A

B={1,2,3,4,5}
2.
Difference: If A and B are two sets, then A Difference B is a set that contains the elements only in A and the elements only in B but not the elements that are in both A and B which can be represented as A-B.
E.g.: A={1,2,3} B={3,4,5} A-B={1,2,4,5}
3.
Intersection: If A and B are two sets, then A intersection B is a set that contains only the elements in both A and B which can be represented as A

B.
E.g.: A={1,2,3} B={3,4,5} A

B={3}.
Disjoint sets: A and B are said to be disjoint sets if they contain no elements in common i.e. A

B=

where

is the Empty set that is the set with no elements.
Add Cartesian product, general union and intersection (i.e. union/intersection of three or more sets -- pp. 8 - 9 of textbook).
Here is a list of some standard set identities:
A, B, C represent arbitrary sets and

is the empty set and U is the Universal Set.
The Commutative laws:
A

B=B

A
A

B=B

A
The Associative laws:
A

(B

C)=(A

B)

C
A

(B

C)=A

B)

C
The Distributive laws:
A

(B

C)=(A

B)

(A

C)
A

(B

C)=(A

B)

(A

C)
The Idempotent laws:
A

A=A
A

A=A
The Absorptive laws:
A

(A

B)=A
A

(A

B)=A
The De Morgan laws:
(A
 B)’=A’  B’
(A
 B)’=A’  B’
Other laws involving Complements :
(A’)’=A
A
 A’= 
A
 A’=U
Other laws involving the empty set :
A

=A
A

=

Other laws involving the Universal Set :

A

U=U
A

U=A
Will include proof for one of the Demorgan’s laws in the Site.
Logic
1.
Proposition and Logical Connections: A “Proposition” can be defined as a declarative statement with sufficient meaning, objective and a definitive truthvalue : true/false .
Connective: Two or more propositions can be combined together to make compound propositions with the help of logical connectives to study express/represent the logical arguments.
The following are the logical connectives used commonly: a. Conjunction:
The logical conjunction is understood in the same way as commonly used “and”.
The compound proposition truth-value is true iff all the propositions hold true. It is represented as “^”.
Truth table for two individual propositions p and q with conjunction is
The truth value of the conjunction of two propositions p and q is defined using the truth table as : p q p^q
T
T
T
F
T
F
F T F
F F F b. Disjunction: This is logical “or” read as either true value of the individual propositions.
Truth table is see conjunction p q pVq
T
T
T
F
T
T
F T T
F F F c. Negation: This is the logical “Negation” read as for  p “not p”.
Truth table is see conjunction

p
T
 p
F
F T d. Conditional: This is used to define as “a proposition holds true” iff another proposition as true i.e. p->q is read as “if p, then q”.
Truth table is see conjunction p q p->q
T T F
T
F
F
T
F
T
F F T e. Biconditional: A proposition (p->q)^(q->p) can be abbreviated using biconditional conjunction <-> as p<->q and is read as “p only if q” and “p if q”.
"p if and only if q" f. Tautology: A compound proposition, which is true in every case.
E.g.: pV
 q p V
 p g. Contradiction: This is the opposite of tautology, which is false in every case.
E.g.: p^
 q p^
 p
Logical implication and equivalence: If the value of Q q is true in every case in, which p is true then p is said to logically imply q, which is represented as p=>q.
If p and q have same truth-value in each case then both are said to be logically
Equivalent represented as p < =>q.
Logical quantifies and quantified statements: Quantifiers are used to make a proposition about a domain individual objects . The proposition variable is bound to a particular domain and not a free variable. Such statements with quantifiers is called a
“quantified statement”.
E.g.: Existential quantifiers

Universal quantifier

Need more on Predicate Logic. See my notes I gave you and also my Web pages on predicate logic.
Need an entry on "Relations" here.
If f is a function and x is an element of its domain, the element of the co-domain associated with x is written as f(x). Move this down below the next paragraph.
A function associates with, or assigns to each element of one set to a single element in the other set. The former set is called the
“domain”
and the later is called “Co-domain” .
It A function is represented as follows:

f : A

B where f is the function, A is the domain and B is the co-domain.
Types of functions:
1.
Onto functions: If A and B are two sets, such that f: A

B and every element in
B is associated with an element in A then f is said to be an onto, surjective or surjection i.e. f(A)=B. Define this notation if you want to use it.
1
2 a b
3
4 c
5
2.
One-to-One functions: If A and B are two sets, such that f: A

B and if no single element of B is associated with more than one element in A, then the function f is said to be One-to-One or injective or an injection.
1
2
3
4 a b c
4 must be mapped to something in B. Otherwise you don't have a function.
3.
Bijection: A function is said to be bijection if it is both Onto and One-to-One.
1
2
3 a b c

Composition of functions:
If f: A

B and g: B

C then, for any element x

A, the function h: A

C defined by h(x)=g(f(x)) is called the composition of g and f and is written as h=g o f.
Inverse of a function:
If f: A

B is a bijection, then for any y

B and x

A such that y=f(x), the inverse of f denoted by f
-1 is defined as x=f
-1
(y).