MMW L#4

Sets

are collections of well defined distinct objects, ideas, or numbers. The groups are called… for as long as the objects in the group share a characteristic and are thus, well defined.

Elements

are objects contained in a set.

Language of Sets

A well-defined set means that it is possible to determine whether an object belongs to a given set. The object that belongs to a set is called members or elements of a the set.

braces { }

A set is always represented by a capital letters and are used to enclose by

∈

We use the symbol … to denote the element of a set. The symbol 1∉B is read as “1 is not an element of set B”

empty or null set

A set which contains no element is called

Venn Diagram

is a diagram that shows all possible logical relations between a finite collection of different sets.

Tabular or roster form

is a method of describing a set where the elements are separated by commas and enclosed by braces

rule form or set builder

notation is a method which makes use of the description 𝑥 ... .“The set of all x such that ...”

Equal sets (=)

Two sets that contain exactly the same elements, regardless of the order listed or possible repetition of elements.

Equivalent sets (~)

Two sets that contain the same number of distinct elements.

Finite sets

A set is finite if the elements that it contains are countable.

Infinite sets

A set is infinite if the elements that it contains are uncountable. The counting of elements has no end.

Cardinality of a set

The number of distinct elements in a set.

Empty set or null set

The set that contains no elements. It can be represented by either or ∅

universal set (U)

is the totality of elements under consideration.

Joint sets

Two sets are joint sets if they have common element/s

Disjoint sets

Two sets are disjoint sets if they have no common element/s.

Subset

For Sets A and B, Set A is a Subset of set B (𝐴 ⊆ 𝐵), if all elements/if every element in Set A belong to Set B.

proper subset ©

For Sets A and B, set A is a Proper Subset of set B if every element in Set A is also in Set B, but Set A does not equal Set B.

The Empty Set is a subset of every Set.

The Empty Set is also a Proper Subset of every Set except the empty set.

Note: ..

number if subsets

2^n

number of proper subsets

2^n - 1

Union of sets A and B

𝐴 ∪ 𝐵 = 𝑠𝑒𝑡 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑓𝑜𝑢𝑛𝑑 𝑜𝑛 𝑏𝑜𝑡h 𝑠𝑒𝑡 𝐴 𝑎𝑛𝑑 𝐵.

Intersection sets of A and B

𝐴 ∩ 𝐵 = 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑐𝑜𝑚𝑚𝑜𝑛 𝑡𝑜 𝑠𝑒𝑡 𝐴 𝑎𝑛𝑑 𝐵.

Compliment of A

𝐴’ = set of all elements in universal set but not found on set A.

Difference of sets A and B

𝐴 − 𝐵=set of all elements in A but not in set B.

Function

can be defined as a rule that relates every element in one set, called the domain, to exactly one element in another set, called the range.

Relation

is any set of ordered-pair numbers. In other words, we can define a relation as a bunch of ordered pairs.

true

All Functions are Relations, but not all Relations are Functions.