Definition of soft set
Understand the definition of soft set and the fundamental operations of soft set. We discuss soft set theory, definition of soft set, parameterization of soft sets, and provide a clear soft set with example to illustrate how this tool handles uncertainty without membership functions.
What is a Soft Set? (Detailed Definition)
A Soft Set is a classification of objects based on some parameters. Think of it as a “parameterized family of subsets.” While a normal set just lists items, a soft set lists items under specific conditions.
2. E = The Parameters (The characteristics or properties).
3. (F, E) = The Soft Set (The mapping of properties to objects).
Example Scenario (The Setup)
Let’s say we are looking for a house to buy.
- Universe (U): {h1, h2, h3, h4} (Four houses available)
- Parameters (E): {e1: Expensive, e2: Beautiful, e3: Near Market}
If a person defines a Soft Set (F, E), they might describe it as:
- F(e1) = {h1, h2} (House 1 and 2 are expensive)
- F(e2) = {h2, h3} (House 2 and 3 are beautiful)
- F(e3) = {h4} (Only house 4 is near the market)
Operations of Soft Sets with Examples
Assume we have two soft sets, (F, A) and (G, B), representing the opinions of two different people, Mr. Smith and Mrs. Smith.
1. Complement of a Soft Set
The complement finds the objects that do not satisfy the parameter.
Complement (Not Expensive):
Fc(e1) = {h3, h4} (Everything in U except h1 and h2).
2. Soft Set Union (OR Operation)
Combining two opinions. If a parameter exists in both sets, we take the union of their elements.
Set B (Mrs. Smith): Beautiful (e2) = {h3, h4}
The Resulting Union:
H(Beautiful) = {h2, h3} ∪ {h3, h4} = {h2, h3, h4}
3. Soft Set Intersection (AND Operation)
Finding common ground. We only look at parameters that both sets have and find the objects they both agree on.
Set B: Near Market (e3) = {h4}
The Resulting Intersection:
H(Near Market) = {h1, h4} ∩ {h4} = {h4}
Soft Numbers and Their Operations
A Soft Number occurs when the “result” of a parameter is not a set of items, but a numerical value (often a fuzzy degree).
| Parameter | Soft Number 1 (S1) | Soft Number 2 (S2) |
|---|---|---|
| Quality | 0.8 | 0.7 |
| Durability | 0.6 | 0.9 |
Arithmetic Example: Addition of Soft Numbers
To add two soft numbers, we add their values parameter by parameter.
2. Durability: 0.6 + 0.9 = 1.5
Resulting Soft Number: {Quality: 1.5, Durability: 1.5}
Arithmetic Example: Multiplication of Soft Numbers
2. Durability: 0.6 × 0.9 = 0.54
Resulting Soft Number: {Quality: 0.56, Durability: 0.54}
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator