1. Sets
- A set is an unordered collection of objects, called elements or members of the set.
- A set is said to contain its elements
if a is element of A
if a is not element of A
- Ways to describe set
- List all the members of a set (Roster method). E.g
(Roster method)
- Use set builder notation:
- List all the members of a set (Roster method). E.g
- A is a subset of B if and only if every element of A is also an element of B.
- Two sets are equal if and only if they have the same elements.
- The empty set:
- Given a set
, the power set of
is the set of all subsets of the set S. The power set of S is denoted by
2. Set operations
- The union of the sets A and B is the set that contains those elements that are either in A or in B, or in both.
- The intersection of the sets A and B is the set containing those elements in both A and B.
- The difference of A and B is the set containing those elements that are in A but not in B. The difference of A and B is also called the complement of B with respect to A.
Table set identies
To be done
3. Cartesian product
- The ordered n-tuple (a1, a2, . . . , an) is the ordered collection that has a1 as its first element, a2 as its second element, . . . , and an an as its nth element.
- The Cartesian product of the sets A1, A2, . . . , An, denoted by A1 x A2 x · · · x An, is the set of ordered n-tuples (a1, a2, . . . , an), where ai belongs to Ai for i = 1, 2, . . . , n.
4. Function
- Let X and Y be nonempty sets. A function f from X to Y is an assignment of exactly one element of X to each element of Y
- A function f is said to be one-to-one, or an injunction, if and only if f(a) = f (b) implies that a = b for all a and b in the domain of f (Injunctive)
- A function f from X to Y is called onto, or a surjection, if and only if for every element y ∈ Y there is an element x ∈ X with f(x) = y (Surjective)
- A function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto. (Bijective)
- Let f be a function from the set X to the set Y and let g be a function from the set y to the set C. The composition of the functions g and f, is defined by is defined by (f ◦ g)(a) = f (g(a)).
5. Cardinality of Sets
Definition
The sets A and B have the same cardinality if and only if there is a one-to-one correspondence from A to B. When A and B have the same cardinality, we write |A| = |B|.
To be done
References
- Discrete Mathematics and Its Applications 7th Edition, Kenneth H. Rosen
- Đại số tuyến tính, Nguyễn Hữu Việt Hưng
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