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}.
Properties of the Complement
Kolman, Bernard. Discrete Mathematical Structures (Pearson Modern Classics for Advanced Mathematics Series) (p. 9). Pearson Education. Kindle Edition.
