DiscreteMath (Midterm - Y1S2)

• It is a well-defined and an unordered collection/aggregate of objects of any kind; the objects are referred as elements, or members of the set

  Set
• It is denoted by upper case/capital letters

  Set
• It is the set that contains all elements relevant to a particular discussion or problem

  Universal Set

  
• x ∈ S: x is an element of S

  x ∉ S: x is NOT an element of S

  Notation
• The number of elements in a set are not countable

  Infinite Set
• The number of elements in a set are countable

  Finite set
• • Roster Method
  • Descriptive or Set Builder Method

  Define/Represent a Set
• It states that the two given sets a re identical, if and only if they contain exactly the same elements

  Set Equality
• Is is a set contained in a larger set or in an equal set

  Subset
• Symbolically: A ⊆ B if ∀x[x ∈ A → x ∈ B] is true


  Subset
• It is a subset that is not equal to the set it belongs to

  Proper Subset
• Symbolically: A ⊂ B if A ⊆ B and A ≠ B

  Proper Subset
• The number of subsets of a set with n elements is 2^n


  Number of Subset
• The number of proper subset of a set with n elements is 2^n-1


  Number of Proper Subset
• It is a way of visually representing sets of items or numbers by using their logical relationships to they should be grouped together

  Venn Diagram
• Set Complement

  
• Set Intersection (∩)

  
• Set Union(∪)

  
• Set Difference (-)

  
• Symmetric Difference (⊕)

  
• It is known as the propositional function or open sentence

  Predicate Logic
• It is a proposition whose TRUTH VALUE depends on the VALUE of the SUBJECT of a statement

  Predicate Logic
• Expressed a P(x)

  Predicate Logic
• It is the set from which the value of the subject x may be chosen for a given P(x)

  Domain/Universe of Discourse
• It expresses the extent to which a predicate P is TRUE (or FALSE) for ALL possible values in the universe of discourse or for SOME value(s) in the universe of discourse

  Quantification
• It is represented by ∀, which means "FOR ALL"

  Universal Quantification
• ∀ x P(x) ↔ P(n1) ^ Р(n2) ^... ^ Р(nk)

  Universal Quantification
• It is represented by ∃, which means "THERE EXISTS"

  Existential Quantification
• ∃ x P(x) ↔ P(n1) v P(n2) v ... v P(nk)

  Existential Quantification
• It is an element/object for which P(x) is FALSE

  Counterexample of ∀xP(x)
• Nested Quantifier

  • It is a quantifier that appears within the scope of another quantifier