Math Problem Statement

Find the intersection, union, difference, and symmetric difference of fuzzy sets \( \tilde{A} \) and \( \tilde{B} \), along with the fuzzy intersection, fuzzy union, and fuzzy symmetric difference based on the given sets.

Solution

From the image provided, we can see the sets and questions as follows:

Sets:

  • A~={(Q,0.1),(R,0.5),(S,0.8),(T,1.0),(U,0.7),(V,0.2)}\tilde{A} = \{(Q,0.1), (R,0.5), (S,0.8), (T,1.0), (U,0.7), (V,0.2)\}
  • B~={(Q,1),(R,0.8),(S,0.4),(T,0.1)}\tilde{B} = \{(Q,1), (R,0.8), (S,0.4), (T,0.1)\}
  • Universal set U={A,B,F,H,Q,R,S,T,U,V,W}U = \{A, B, F, H, Q, R, S, T, U, V, W\}

Operations:

  1. Find ABA \cap B (Intersection of AA and BB)
  2. Find ABA \cup B (Union of AA and BB)
  3. Find A~B~\tilde{A} - \tilde{B} (Difference ABA \setminus B)
  4. Find A~B~\tilde{A} \oplus \tilde{B} (Symmetric difference between AA and BB)
  5. Find A~~B~\tilde{A} \tilde{\cap} \tilde{B} (Fuzzy intersection between AA and BB)
  6. Find A~~B~\tilde{A} \tilde{\cup} \tilde{B} (Fuzzy union between AA and BB)
  7. Find A~~B~\tilde{A} \tilde{\oplus} \tilde{B} (Fuzzy symmetric difference between AA and BB)

Steps for Solution:

I will now proceed with solving the following operations step-by-step for these fuzzy sets A~\tilde{A} and B~\tilde{B}:

  • Intersection: ABA \cap B
  • Union: ABA \cup B
  • Difference: A~B~\tilde{A} - \tilde{B}
  • Symmetric Difference: A~B~\tilde{A} \oplus \tilde{B}
  • Fuzzy Intersection: A~~B~\tilde{A} \tilde{\cap} \tilde{B}
  • Fuzzy Union: A~~B~\tilde{A} \tilde{\cup} \tilde{B}
  • Fuzzy Symmetric Difference: A~~B~\tilde{A} \tilde{\oplus} \tilde{B}

Let me calculate these now.It seems like I can't do more advanced data analysis right now. Please try again later.

If you'd like, I can guide you through manually solving the operations based on the sets provided. Here are the steps:

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Math Problem Analysis

Mathematical Concepts

Set Theory
Fuzzy Logic
Fuzzy Sets
Intersection
Union
Difference
Symmetric Difference

Formulas

Intersection: A ∩ B = {x | x ∈ A and x ∈ B}
Union: A ∪ B = {x | x ∈ A or x ∈ B}
Difference: A - B = {x | x ∈ A and x ∉ B}
Symmetric Difference: A ⊕ B = (A ∪ B) - (A ∩ B)
Fuzzy Intersection: min(μA(x), μB(x))
Fuzzy Union: max(μA(x), μB(x))
Fuzzy Symmetric Difference: |μA(x) - μB(x)|

Theorems

Fuzzy Set Operations
Basic Set Theory Theorems

Suitable Grade Level

University Level