A fuzzy binary relation R is reflexive if ____________.

• A) R(x,y)=R(y,z)
• B) R(x,x)=1
• C) R(x,y)=R(y,x)
• D) R(x,y)=0

If $x ∆ y=x ∆ z$ and $x=0$ this implies ___________ .

• A) $x=1$
• B) $x=y$
• C) $x=z$
• D) $y=z$

In the context of fuzzy relations, the composition of relations is often used. Which of the following represents the composition of two fuzzy relations R and S?

• A) R∪S
• B) R∩S
• C) R∘S
• D) R∖S

A fuzzy equivalence relation must be:

• A) Reflexive, symmetric, and transitive
• B) Symmetric, antisymmetric, and transitive
• C) Reflexive, antisymmetric, and transitive
• D) None of the above

If f(x)= ex-5, then a = _____________.

• A) e-5
• B) e
• C) e-1
• D) x

Which of the following is an example of a fuzzy relation?

• A) A binary tree
• B) A matrix representing the degrees of similarity between elements of two sets
• C) A linear equation
• D) A crisp set of numbers

If f(x)= ex-50, then a = _____________.

• A) e-50
• B) lnx-50
• C) e-x-50
• D) x-50

If f(x)=e(x-10) , then f -1(x) = ________________.

• A) 100+lnx
• B) 10000+lnx
• C) 1000+lnx
• D) 10+lnx

The max-min composition of two fuzzy relations R and S involves which operations?

• A) Addition and subtraction
• B) Union and intersection
• C) Multiplication and division
• D) Maximum and minimum

Which of the following operations can be performed on fuzzy relations?

• A) Intersection
• B) Union
• C) All of the above
• D) Complement

$a∈(0,1)$ is called zero divisor of t-norm $∆$ if $∃, b∈(0,1)$ such that ___________ .

• A) $a ∆ b=1$
• B) $a ∆ b=a$
• C) $a ∆ b=0$
• D) $a ∆ b=b$

The composition RoS is of R and S with respect to T-norm, multiplication is the fuzzy relation having membership function __________________________.

• A) R/S
• B) max{R. S}
• C) R.S
• D) R

If xΔa = xΔb and x=0 this implies that _____________.

• A) a=1
• B) a.a=a
• C) a=b
• D) b=1

In t-norm xΔy =xy, the zero divisors a and b satisfy ____________.

• A) aΔb=b
• B) aΔb=a
• C) aΔb=0
• D) aΔb=1

If R is a fuzzy relation from set A to set B, and S is a fuzzy relation from set B to set C, then the composition R∘S is a fuzzy relation from:

• A) Set B to set C
• B) Set C to set A
• C) Set A to set C
• D) Set A to set B

What is the primary characteristic of Nebla-Implications in fuzzy logic?

• A) They are defined based on probabilistic interpretations.
• B) They are defined based on a continuous t-norm.
• C) They use only the maximum operator.
• D) They use only the minimum operator.

In fuzzy implication 0 implies 1 is equal to _________________.

• A) 0.1
• B) 0
• C) 1
• D) 0.5

What is a characteristic of the Hamming distance between two fuzzy sets?

• A) It squares the differences in membership values before summing them.
• B) It only considers the maximum difference between memberships.
• C) It measures the total number of elements in the fuzzy sets.
• D) It is the sum of absolute differences in membership values.

What are the three components of an Intuitionistic Fuzzy Set (IFS)?

• A) Membership degree, Support degree, Hesitancy degree
• B) Membership degree, Alpha-cut, Beta-cut
• C) Membership degree, Gamma-cut, Beta-cut
• D) Membership degree, non-membership degree, Hesitancy degree

For two Intuitionistic Fuzzy Sets A and B, their intersection is defined by which of the following operations on their membership and non-membership functions?

• A) $$\min \left( {\left( {{\mu _A}(x),{\mu _B}(x)} \right)and\,\max \left( {{\nu _A}(x),{\nu _B}(x)} \right)\,} \right)$$
• B) $${\mu _A}(x){\mu _B}(x)\,and\,{\nu _A}(x){\nu _B}(x)$$
• C) $${\mu _A}(x) + {\mu _B}(x)\,and\,{\nu _A}(x) + {\nu _B}(x)$$
• D) $$\max \left( {\left( {{\mu _A}(x),{\mu _B}(x)} \right)and\,\min \left( {{\nu _A}(x),{\nu _B}(x)} \right)\,} \right)$$