# Fuzzy properties, De Morgan laws and α-cuts

Subject: Intelligent Systems II

Exam: M2

Reader: https://deep-learning-study-room.vercel.app/is2/#m2/read/fuzzy-core/properties

Canonical: https://deep-learning-study-room.vercel.app/read/is2/m2/read/fuzzy-core--properties/

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.