Intelligent Systems II · M2 · read

Fuzzy properties, De Morgan laws and α-cuts

2. Fuzzy logic, from meaning to full solutions

Definitions → membership graphs → operations → rules → defuzzification → complete controller.

Translate the property names

Read this	Meaning	Concrete example
Commutative	Changing order changes nothing	max(0.3,0.8)=max(0.8,0.3)=0.8
Associative	Regroup the same operation	max(max(0.2,0.7),0.4)=max(0.2,max(0.7,0.4))=0.7
Idempotent	Combining a membership with itself keeps it	min(0.6,0.6)=0.6
Involution	Complement twice returns the original	1−(1−0.4)=1−0.6=0.4
De Morgan	NOT swaps AND and OR while complementing each operand	1−max(a,b)=min(1−a,1−b)

Properties that remain valid

Property	Statement for max–min operations
Commutative	A ∪ B = B ∪ A; A ∩ B = B ∩ A
Associative	(A ∪ B) ∪ C = A ∪ (B ∪ C); likewise for intersection
Idempotent	A ∪ A = A; A ∩ A = A
Distributive	A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C); and its dual
Involution	(Aᶜ)ᶜ = A
De Morgan	(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ; (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ

Two crisp identities do not generally hold

For fuzzy sets, A ∪ Aᶜ need not equal the entire universe at membership 1; A ∩ Aᶜ need not be the empty set at membership 0.

At an element x, μA(x)=0.4. Calculate μ(A ∪ Aᶜ) and μ(A ∩ Aᶜ). [2 marks · Practice]

Given

μA(x)=0.4; standard complement/max/min.

To find

Whether excluded middle and non-contradiction hold at this element.

Formula and why it applies

Complement first, then use max or min.

Solution

Step 1. Complement

\mu_{A^c}(x)=1-0.4=0.6

Step 2. Union

\max(0.4,0.6)=0.6\ne1

Step 3. Intersection

\min(0.4,0.6)=0.4\ne0

Final answer. The union is 0.6 and the intersection is 0.4; the crisp identities fail here.

Verify the first De Morgan law for μA=0.3 and μB=0.8. [2 marks · Practice]

Given

A=0.3, B=0.8 at the same element.

To find

Both sides of (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ.

Formula and why it applies

Evaluate the operations independently on each side.

Solution

Step 1. Left side

1-\max(0.3,0.8)=1-0.8=0.2

Step 2. Right side

\min(1-0.3,1-0.8)=\min(0.7,0.2)=0.2

Final answer. Both sides equal 0.2.

For U={a,b,c,d} and memberships A=[0,0.3,1,0.7], find support, core, boundary, height and the 0.7-cut. [2 marks · Worked example]

Given

Universe U={a,b,c,d}. Memberships: A(a)=0, A(b)=0.3, A(c)=1, A(d)=0.7. Cut level α=0.7.

To find

The support, core, boundary and 0.7-cut as sets of elements, and the height as one membership value.

Formula and why it applies

Support: degree>0; core: degree=1; boundary: 0<degree<1; height: largest degree; α-cut: degree≥α.

Solution

Step 1. Inspect each element

Element	Degree	Positive?	Exactly 1?	At least 0.7?
a	0	No	No	No
b	0.3	Yes	No	No
c	1	Yes	Yes	Yes
d	0.7	Yes	No	Yes

Step 2. Collect the requested elements

Support={b,c,d}. Core={c}. Boundary={b,d}. The 0.7-cut={c,d}.

Step 3. Read the height

The maximum degree is 1, so the height is 1 and the set is normal.

Final answer. Support {b,c,d}; core {c}; boundary {b,d}; height 1; 0.7-cut {c,d}.

Find the 0.5-cut of the triangular fuzzy set Tri(2,5,8). [2 marks · Worked example]

Given

Triangle start a=2, peak b=5, end c=8; cut level α=0.5. Retain all inputs whose membership is at least 0.5.

To find

Left and right endpoints of the retained interval.

Formula and why it applies

The triangle rises past 0.5, remains above it, then falls through 0.5. Solve height=0.5 on both slopes.

Solution

Step 1. Left slope: multiply both sides by 3

(x-2)/3=0.5

x-2=0.5\times3=1.5

Step 2. Isolate the left input

x=1.5+2=3.5

Step 3. Right slope: multiply both sides by 3

(8-x)/3=0.5

8-x=1.5

Step 4. Isolate the right input

x=8-1.5=6.5

Step 5. Include all inputs between them

The graph is at or above 0.5 for 3.5≤x≤6.5. Include the endpoints because the condition is ≥.

Final answer. The 0.5-cut is [3.5,6.5].

Describing a fuzzy set

Support: inputs with μ > 0. Core: inputs with μ = 1. Boundary: inputs with 0 < μ < 1. Height: largest membership (more generally the supremum). A set is normal when its height is 1; it is subnormal when its height is below 1. A crossover point has membership 0.5. An α-cut contains inputs with μ ≥ α.

For Tri(2,5,8), support is (2,8), core is {5}, height is 1, and the 0.5-cut is [3.5,6.5]. Solve (x−2)/3=0.5 → x=3.5 and (8−x)/3=0.5 → x=6.5.

A one-dimensional fuzzy set is convex if every α-cut is an interval (or empty). The membership between two inputs must not dip below the smaller endpoint membership. A single triangle is convex; two separated peaks with a lower valley may not be.

Yaad rakho: Fuzzy A aur NOT A dono partial ho sakte hain. Isliye crisp waali 1 aur 0 identities blindly nahi lagani.