Fuzzy set operations

Key takeaways

Fuzzy set operations belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Fuzzy set operations to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Fuzzy set operations from memory before moving on to harder problems.

Reference excerpt

Fuzzy set operations are a generalization of crisp set operations for fuzzy sets. There is in fact more than one possible generalization. The most widely used operations are called standard fuzzy set operations; they comprise: fuzzy complements, fuzzy intersections, and fuzzy unions.

Standard fuzzy set operations Let A and B be fuzzy sets that A,B ⊆ U, u is any element (e.g. value) in the U universe: u ∈ U.

Standard complement

μ ¬ A ( u ) = 1 − μ A ( u ) {\displaystyle \mu _{\lnot {A}}(u)=1-\mu _{A}(u)}

The complement is sometimes denoted by ∁A or A∁ instead of ¬A.

Standard intersection

μ A ∩ B ( u ) = min { μ A ( u ) , μ B ( u ) } {\displaystyle \mu _{A\cap B}(u)=\min\{\mu _{A}(u),\mu _{B}(u)\}}

Standard union

μ A ∪ B ( u ) = max { μ A ( u ) , μ B ( u ) } {\displaystyle \mu _{A\cup B}(u)=\max\{\mu _{A}(u),\mu _{B}(u)\}}

In general, the triple (i,u,n) is called De Morgan Triplet iff

i is a t-norm, u is a t-conorm (a.k.a. s-norm), n is a strong negator, so that for all x,y ∈ [0, 1] the following holds true:

u(x,y) = n( i( n(x), n(y) ) ) (generalized De Morgan relation). This implies the axioms provided below in detail.

Fuzzy complements μA(x) is defined as the degree to which x belongs to A. Let ∁A denote a fuzzy complement of A of type c. Then μ∁A(x) is the degree to which x belongs to ∁A, and the degree to which x does not belong to A. (μA(x) is therefore the degree to which x does not belong to ∁A.) Let a complement ∁A be defined by a function

c : [0,1] → [0,1] For all x ∈ U: μ∁A(x) = c(μA(x))

Axioms for fuzzy complements Axiom c1. Boundary condition c(0) = 1 and c(1) = 0 Axiom c2. Monotonicity For all a, b ∈ [0, 1], if a < b, then c(a) > c(b) Axiom c3. Continuity c is continuous function. Axiom c4. Involutions c is an involution, which means that c(c(a)) = a for each a ∈ [0,1] c is a strong negator (a.k.a. fuzzy complement). A function c satisfying axioms c1 and c3 has at least one fixpoint a* with c(a*) = a*, and if axiom c2 is fulfilled as well there is exactly one such fixpoint. For the standard negator c(x) = 1-x the unique fixpoint is a* = 0.5 .

Fuzzy intersections

The intersection of two fuzzy sets A and B is specified in general by a binary operation on the unit interval, a function of the form

i:[0,1]×[0,1] → [0,1]. For all x ∈ U: μA ∩ B(x) = i[μA(x), μB(x)].

Axioms for fuzzy intersection Axiom i1. Boundary condition i(a, 1) = a Axiom i2. Monotonicity b ≤ d implies i(a, b) ≤ i(a, d) Axiom i3. Commutativity i(a, b) = i(b, a) Axiom i4. Associativity i(a, i(b, d)) = i(i(a, b), d) Axiom i5. Continuity i is a continuous function Axiom i6. Subidempotency i(a, a) < a for all 0 < a < 1 Axiom i7. Strict monotonicity i (a1, b1) < i (a2, b2) if a1 < a2 and b1 < b2 Axioms i1 up to i4 define a t-norm (a.k.a. fuzzy intersection). The standard t-norm min is the only idempotent t-norm (that is, i (a1, a1) = a for all a ∈ [0,1]).

Fuzzy unions The union of two fuzzy sets A and B is specified in general by a binary operation on the unit interval function of the form

u:[0,1]×[0,1] → [0,1]. For all x ∈ U: μA ∪ B(x) = u[μA(x), μB(x)].

Axioms for fuzzy union Axiom u1. Boundary condition u(a, 0) =u(0 ,a) = a Axiom u2. Monotonicity b ≤ d implies u(a, b) ≤ u(a, d) Axiom u3. Commutativity u(a, b) = u(b, a) Axiom u4. Associativity u(a, u(b, d)) = u(u(a, b), d) Axiom u5. Continuity u is a continuous function Axiom u6. Superidempotency u(a, a) > a for all 0 < a < 1 Axiom u7. Strict monotonicity a1 < a2 and b1 < b2 implies u(a1, b1) < u(a2, b2) Axioms u1 up to u4 define a t-conorm (a.k.a. s-norm or fuzzy union). The standard t-conorm max is the only idempotent t-conorm (i. e. u (a1, a1) = a for all a ∈ [0,1]).

Aggregation operations Aggregation operations on fuzzy sets are operations by which several fuzzy sets are combined in a desirable way to produce a single fuzzy set. Aggregation operation on n fuzzy set (2 ≤ n) is defined by a function

h:[0,1]n → [0,1]

Axioms for aggregation operations fuzzy sets Axiom h1. Boundary condition h(0, 0, ..., 0) = 0 and h(1, 1, ..., 1) = one Axiom h2. Monotonicity For any pair <a1, a2, ..., an> and <b1, b2, ..., bn> of n-tuples such that ai, bi ∈ [0,1] for all i ∈ Nn, if ai ≤ bi for all i ∈ Nn, then h(a1, a2, ...,an) ≤ h(b1, b2, ..., bn); that is, h is monotonic increasing in all its arguments. Axiom h3. Continuity h is a continuous function.

See also Fuzzy logic Fuzzy set T-norm Type-2 fuzzy sets and systems De Morgan algebra

Further reading Klir, George J.; Bo Yuan (1995). Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall. ISBN 978-0131011717.

References

L.A. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965

Worked examples

Example 1 — a first encounter with Fuzzy set operations

Start with the simplest possible case. Write down what Fuzzy set operations claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Fuzzy set operations before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Fuzzy set operations ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Fuzzy set operations

In research: Fuzzy set operations appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Fuzzy set operations in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Fuzzy set operations is common in secondary-school and first-year university syllabi. It links to neighbouring topics Fuzzy logic, so understanding it makes those chapters shorter.
In everyday life: Look for Fuzzy set operations outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Fuzzy set operations in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Fuzzy set operations means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Fuzzy set operations out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Fuzzy set operations in simple terms?

Fuzzy set operations are a generalization of crisp set operations for fuzzy sets. There is in fact more than one possible generalization.

Why does Fuzzy set operations matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Fuzzy set operations?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Fuzzy set operations.