Sets and their Representation

Sets and their Representation

• Sets are collections of well-defined objects, usually represented by capital letters.
• Sets are denoted by braces and commas, containing their elements.
• An object belonging to a set is called a member or element, denoted by the symbol ∈.
• A set is well-defined if we know exactly if an object belongs to it.
• A variation of notation using ellipses (...) can be used to describe large sets, indicating "and so forth".
• Sets are denoted by braces {}, commas ,, and the symbol ∈ which means "is an element of".

Describing Sets

• Listing Method/ Roster Method: listing all elements enclosed in braces.
• Rule Method/ Set-Builder Notation: descriptive phrase in braces, defining set elements. Example: A = {x/x is an even number less than 5}
• Interval Notation: uses brackets [] and parentheses () to represent ranges on a number line. Square brackets indicate inclusion (≤), and parentheses indicate exclusion (<).

Types of Sets

• Empty Set: contains no members, denoted by {} or the Danish letter ∅.
• Finite Set: has a countable number of elements, meaning you can list all elements.
• Infinite Set: has an uncountable number of elements, impossible to list all elements.

Cardinality of a Set

• Cardinality: the number of elements in a set, denoted by |A| or n(A).
• Example: If F = {1, 2, 3, 4, 5}, then n(F) = 5.

Equal and Equivalent Sets

• Equal Sets: have exactly the same elements.
• Equivalent Sets: have the same number of elements.