Foundational Definitions

• Set: A collection of elements; symbolized with ‘{...}‘; e.g. (X) = {1, 3, 5, 8}, (Y) = {5, 8, 10, 35}, (Z) = {8, 10, 36, 37}
• Element of: An element is a member of a set; symbolized with ‘∈‘; e.g. 3 ∈ {1, 3, 5, 8} or b ∈ B
• Not an element of: An element is not a member of a set; symbolized with ‘∉‘; e.g. 7 ∉ {1, 3, 5, 8}
• Null Set (or Empty Set): A set with no elements; symbolized with ‘{}‘ or ‘Ø‘;
• Cardinality: Number of elements of a set; symbolized with |A| or #A; e.g. |X| = 4 (see Set for definition of X)
• Set Builder Notation: A = {x : x ∈ ℕ} the set A is equal to a set of x such that (‘|’ can also be used) x are elements of ℕ
• Ordered Pair / Tuple: An ordered collection of elements, where order matters (unlike a set); symbolized with (a, b); e.g. (1, 2) ≠ (2, 1), whereas {1, 2} = {2, 1}

Logical Quantifiers

• Universal Quantifier: For all of; symbolized with ‘∀’; e.g. ∀ x, P(x) means P(x) is true for all of x
• Existential Quantifier: There exists at least one; symbolized with ‘∃’; e.g. ∃ x : P(x) means there exists at least one x such that P(x) is true
• Uniqueness Quantifier: There exists exactly one; symbolized with ‘∃!’; e.g. ∃! x : P(x) means there exists exactly one x such that P(x) is true

Logical Connectives

• Negation ¬ “not”
• Conjunction ∧ “and”
• Disjunction ∨ “or”
• Implication ⟹ or → “if…then”
• Biconditional ⇔ “if and only if”

Relations between Sets

• Equality: Elements of one set are equal to the other set; symbolized with ‘=’; e.g. {1, 2} = {1, 2}
• Equivalency: Elements of one set are equivalent to the other set; symbolized with ‘≡‘; A ≡ B
• Subset: Every element of set A is also an element of set B, where A ⊆ B; symbolized with ‘⊆’; e.g. {5} ⊆ {1, 3, 5, 8, 10, 35} or {1, 2} ⊆ {1, 2}
• Proper Subset: Every element of set A is also an element of set B but both sets do not have equality, where A ⊂ B; symbolized with ‘⊂’; e.g. {5} ⊂ {1, 3, 5, 8, 10, 35}
• Not a Proper Subset: Every element one set is not an element of another set; symbolized with ‘⊄’; e.g. A ⊄ B
• Superset: Every element of set A is an element of set B, where B ⊇ A; symbolized with ‘⊇’; {1, 3, 5, 8, 10, 35} ⊇ {5} or {1, 2} ⊇ {1, 2}
• Proper Superset: Every element of B is also an element of A but B ≠ A; symbolized ‘⊃’; e.g. {1, 3, 5, 8, 10, 35} ⊃ {5}
• Disjoint Sets: Two sets that share no elements; expressed as A ∩ B = ∅; Example: {1, 2} and {3, 4} are disjoint
• Partition: A way of splitting a set into non-overlapping, non-empty subsets whose union is the whole set; often symbolized with {Aᵢ}; Example: {1,2,3,4} can be partitioned into {1,2} and {3,4}

Operations on Sets

• Union: Union of two sets means the summation of all elements of those sets; symbolized with ‘∪’; e.g. X ∪ Y = {1, 3, 5, 8, 10, 35} (see Set for definition of X and Y)
• Disjoint Union: Union of sets that keeps track of which set each element came from, even if the sets overlap; symbolized with ⊔ or ⊎
• Intersection: Intersection of two sets means the resultant set only includes elements that belong to both sets; symbolized with ‘∩’; e.g. X ∩ Y = {5, 8} (see Set for definition of X and Y)
• Indexed Union / Indexed Intersection: Union or intersection over a whole family of sets at once, not just two; symbolized with ⋃ᵢ ∈ I Aᵢ and ⋂ᵢ ∈ I Aᵢ; Example: ⋃ᵢ∈{1,2,3} Aᵢ = A₁ ∪ A₂ ∪ A₃
• Relative Complement (Difference): Elements that are members to A but not members to B; symbolized with A \ B or A – B; e.g. X – Y = {1, 3} (see Set for definition of X and Y)
• Complement: Every element in Universal set but not in A, were A’ or A^c or Ā; e.g. X^c = {10, 35, 36, 37} (see Set for definition of X, Y and Z)
• Symmetric Difference: Element that are in A or B but not members of the intersection of A and B; symbolized with ‘Δ’; e.g. X Δ Y = {1, 3, 10, 35} (see Set for definition of X and Y)
• Power Set: Given a set A, a power set of A is the set of all subsets of the set; symbolized with ‘P(A)’; e.g. {a, b} ∈ A, P(A) = {Ø, {a}, {b}, {a,b}}
• Cartesian Product: Set of all ordered pairs from A and B; symbolized with A × B; e.g. {1,2} × {3,4} = {(1,3), (1,4), (2,3), (2,4)}
• Indexed Cartesian Product: The set of all possible tuples formed by picking one element from each set in an indexed family; symbolized with ∏ᵢ∈I Vᵢ; e.g. Let V₁ = {a}, V₂ = {1, 2}, V₃ = {x, y, z}, therefore ∏ᵢ∈{1,2,3} Vᵢ = {(a, 1, x), (a, 1, y), (a, 1, z),(a, 2, x), (a, 2, y), (a, 2, z)}. This generalizes the two-set case (A × B) to any number of sets, indexed by a set I.

Standard Number Sets

• Natural numbers: ℕ; positive whole numbers
• Integers: ℤ; negative or positive whole numbers and zero
• Rational Numbers: ℚ
• Real numbers: ℝ; ℝ = {x : -∞ < x < ∞}
• Complex Numbers: ℂ
• Aleph-null (ℵ₀) : The cardinality of the natural numbers (the smallest infinite cardinal);
• Universal Set: The set that contains all elements under consideration for a particular discussion or problem, including all its subsets; symbolized with U (sometimes Ω or 𝕌); e.g. if U = {1, 2, 3, …, 10} and A = {2, 4, 6}, then Aᶜ = U \ A = {1, 3, 5, 7, 8, 9, 10}

Functions and Mappings

• to: Denotes a function’s domain and codomain; symbolized with ‘→’; e.g. f : X → Y means function f maps the set X (domain) into set Y (codomain)
• maps to: Denotes where a specific element is sent; symbolized with ‘↦’; e.g. f : x ↦ y means function f maps the element x to the element y

Zermelo-Fraenkel set theory (or ZFC) is nine axioms that define Set Theory. These nine axioms are provided below. Note that if you’re interested in a more formal (but understandable) description of these axioms then visit here:

Axiom of Extensionality

Translation: If two sets have the same elements then the sets are equal to each other.

Implications to Systems: If two components expose identical sets of inputs, outputs and behaviors, Extensionality licenses treating them as being equal. This is true even if their implementation is completely different and thus the Axiom of Extensionality underpins “black box” abstraction in Systems Engineering.

Likewise if two systems satisfy the same at of requirements then, from a verification of requirement perspective they are the same.

Axiom of Pairing

Translation: Given any two elements, there’s a set that contains precisely those two elements.

Implications to Systems: This axiom by itself seems trivial but it allows for the definition of ordered pairs which is very much relevant to Systems Engineering. See Kuratowski’s Definition for Ordered Pairs.

Axiom Schema of Comprehension (or Separation)

Translation: A subset of a set can be created by a rule.

Implication to Systems: This axiom is the foundation to defining the boundary of the system according to a rule, or defining boundaries of respective sub-systems after decomposition. What’s important here it that the rule is applied consistently across the universal set.

Axiom of Union

Translation: For any set of sets, there exists a set containing exactly the elements that belong to at least one of those sets.

Implication to Systems: It guarantees that, no matter how a system is broken into subsystems, recombining their elements always produces one comprehensive, well-defined set.

Axiom of Power Set

Translation: For a given set there exists a power set.

Implication to Systems: For a system with n binary state variables the possible configurations of the system is 2ⁿ. This rapid explosion of possible configurations is why decomposition of a system is desirable – to minimize the state space of the decomposed system partition.

Axiom of Infinity

Translation: A set with infinite number of elements exists. It draws the distinction with potential infinity (adding a number each time which is a never-ending process) and actual infinity which you can point to as a single object.

Implication to Systems: This axiom allows for the simulation of both discrete-time systems and continuous-time systems.

Axiom of Replacement

Translation: A function takes any set A to a new set B.

Implication to Systems: The clearest example of this is mapping from one representation, or viewpoint, to another representation, i.e. mapping the functional architecture to physical architecture, or requirements model to the system model.

Axiom of Regularity

Translation: A set can not have an element of itself.

Implication to Systems: This axiom avoids infinite regression of a system hierarchy, which would be the case if it was said that the sub-system contains itself.

Axiom of Choice

Translation: Given any collection of nonempty sets, exactly one element from each of them can be chosen simultaneously, even if there are infinitely many sets and no explicit rule for choosing.

Implication to Systems: When you want to choose test case, one requirement, one component of a set of such elements you’re informally leaning on the Axiom of Choice.

Kuratowski uses ZFC’s Axiom of Pairing and ZFC’s Axiom of Extensionality to define ordered pairs. Why are ordered pairs a big deal in Systems Engineering? Each time a directional relationship is created – think causal links to represent functional flows or information flows or creating a link between a requirement and its verification method – the author of that relationship is actually using an ordered pair. Ordered pairs are special because, as the name suggests, there is an order to the elements. An ordered pair is represented in set theory as (a, b) which says that element a comes first then element b comes after. A particular instance may be referred to as a relationship. This is a subtle distinction with the set {a, b} which is not ordered and indeed {a, b} is {b, a} because of the Axiom of Extensionality.

Kuratowski defines an ordered pair as

(a, b) = {{a},{a,b}}

Kuratowski uses Axiom of Pairing three times:

1. {a, a} is a pairing of a with itself. It is the same as {a}
2. Pair a now with b to form {a, b}. Note this is a set and is not ordered.
3. Now pair {a} with {a, b} to form {{a}, {a, b}} which is equivalent to (a, b).

This is ordered because if we wanted to have (b, a) then we would end up with {{b}, {a, b}} which is of course not the same as {{a}, {a, b}} when a ≠ b (which is where the Axiom of Extensionality comes in).

See here for a more detailed description on this definition.