Sets

• A set is an unordered collection of objects called elements or members
• (a+b)(c+d) = ac + ad + bc + bd
• Sets are one of the basic building blocks of programming languages. These are also importance in counting.
• The study about sets is embodied in Mathematics called, "Set Theory."
• The founder of set theory is Georg Countor suggested that imagining set is an unordered collection of objects, so the order does not matter.
• The objects in a set are called elements, or members of the set.
• A set can be represented by listing its elements between curly braces { } separated by commas.
• If we have a finite number of elements, then it's called as Finite Set.
• Empty Set - when there is nothing inside the set.
• Infinite Set - if there is no end to the number of elements.
• The notation a ∈ A denotes that a is an element of the set A.
• The notation a ∉ A denotes that a is not an element of the set A.
• In roster method, all of the elements are explicitly stated, and the order of the elements or members is not important.
• Listing a member or element does not change the set.
• When a pattern is clear in a set, Elipses (...) can be used to describe when the pattern is clear.
• In order to list the members of a set, we use curly {} brackets to denote inclusion in a set.

  All members in a set are inside the brackets, and any objects outside the bracket are not members of a set.
• Another way to describe a set is Set-Builder Notation.
• Venn diagrams show the relationship between two or more sets by drawing circles representing those sets and showing where they intersect.
• Set Builder Notation uses a colon : to separate the defining property from the set itself.
• A predicate may be used when defining Set-Builder Notations:

  For example,

  • S = {x | P(x)} - an example is S = {x | Prime(x)} which means S is composed of x such that prime x. Meaning, only prime numbers are included.
• Complement of a set A is denoted as A'
• The union of two sets A and B is denoted as A U B
• Another way of describing a set is through interval notation.
• These are some of important sets:

  • R = set of real numbers
  • R+ = set of positive real numbers
  • C = set of complex numbers
  • Q = set of rational numbers
• An example of difference between roster notation, set-builder notation, and interval notation:

  • Interval Notation: [1, 7]
  • Roster Notation: {1, 2, 3, 4, 5, 6, 7}
  • Set-builder: = {x | 1 ≤ x ≤ 7}
• Difference between interval and set-builder notation forms.

  
• Closed interval: [a, b]

  Open interval: (a, b)
• The universal set U is the set containing everything currently under consideration. It is sometimes, implicit, explicit, and its contents depend on the context.
• The empty set is the set with no elements which is symbolized as ∅ or {}.
• Venn diagram is a pictorial representation of a set.
• The venn diagram is introduced by John Venn.
• Sets can be elements of set.
• The empty set is different from a set containing the empty set. True or False?


  True. Ø is not equal to {Ø}. Ø means that the set is empty or has no elements.
• Sets are equal if and only if they have the same elements. True or False?

  True. This is also known as set equality. Example:

  
• What is the definition of subset?


  The set A is a subset of B, if and only if every element of A is also an element of B. The notation A ⊆ B is used to indicate that A is a subset of the set B.

  
• What does the quantifier (∀x) mean?

  The symbol (∀x) is a universal quantifier in mathematical logic. It is read as "for all" or "for every." When you see (∀x) in a mathematical statement, it indicates that the statement following it applies to every element in a certain set or domain.

  
• Proper subset means that every element in A is found in B, but A is not equal to B.

  
• Proper subset means that ∀x ∧ (x∈A -> x ∈ B) ∧ ( x ∃x(x∈B∧x∈/A)

  
• Set Cardinality refers to distinct elements in a set.
• What is the definition of power sets?

  The definition of a power set is the subsets of a set A. It is denoted as P(A), and is called the power set of A.

  The power of a set is denoted as 2^n, where n is the cardinality of a set.