Ch. 1: Sets, sequences, matrices Flashcards
z+
{x | x is a positive integer} (1,2,3…)
N
{x | x is a positive integer or zero} (Natural number)
Z
{x | x is an integer} (-2, -1, 0, 1, 2…)
Q
{x | x is a rational number} (a/b where a and b are integers and b is not zero)
R
{x | x is a real number} ( real number is a value of a continuous quantity that can represent a distance along a line. … The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2)
Empty set
Denoted by { } or symbol ∅
sets A and B are equal if
if they have the same elements, and we write A = B.
Subsets
If every element of A is also an element of B, that is, if whenever x∈A then x∈B, we say that A is a subset of B or that A is contained in B, and we write A⊆B. If A is not a subset of B, we write A B.
Let A be a set and let B = {A,{A}}.
Then, since A and {A} are elements of B, we have A∈B and {A} ∈B. It follows that {A} ⊆B and {{A}} ⊆B. However, it is not true that A⊆B.
Empty subsets
For any set A, since there are no elements of∅that are not in A, we have ∅⊆A. Note: { } is not always an element of A
Equality as denoted by subsets
It is easy to see that A = B if and only if A⊆B and B⊆A. This simple statement is the basis for proofs of many statements about sets.
Union
A∪B = {x | x∈A or x∈B}. (or logic)
Intersection
A∩B = {x | x∈A and x∈B}. (and logic)
Disjoint set
Two sets that have no common elements,
complement of B with respect to A
the set of all elements that belong to A but not to B, A−B. A−B = {x | x∈A and x∈/ B}.
If U is a universal set containing A, then U−A is called the complement of A
denoted by A. Thus A = {x | x∈/ A}.
symmetric difference
the set of all elements that belong to A or to B, but not to both A and B, and we denote it by A⊕B. Thus A⊕B = {x | (x∈A and x∈/ B) or (x∈B and x∈/ A)}.
If U is a universal set containing A, then U−A is called the complement of A
denoted by A(hat). Thus A(hat) = {x | x∈/ A}.
Commutative Properties
- A∪B = B∪A 2. A∩B = B∩A
Associative Properties
- A∪(B∪C) = (A∪B)∪C 4. A∩(B∩C) = (A∩B)∩C
Distributive Properties
- A∩(B∪C) = (A∩B)∪(A∩C) 6. A∪(B∩C) = (A∪B)∩(A∪C) Kolman, Bernard. Discrete Mathematical Structures (Pearson Modern Classics for Advanced Mathematics Series) (p. 8). Pearson Education. Kindle Edition.
Idempotent Properties
- A∪A = A 8. A∩A = A Kolman, Bernard. Discrete Mathematical Structures (Pearson Modern Classics for Advanced Mathematics Series) (p. 9). Pearson Education. Kindle Edition.
Properties of the Complement
Kolman, Bernard. Discrete Mathematical Structures (Pearson Modern Classics for Advanced Mathematics Series) (p. 9). Pearson Education. Kindle Edition.
