The Neutrosophic Multisets and the Neutrosophic Multiset Algebraic Structures were introduced

by Florentin Smarandache [2, 3, 4] in 2016.

1. Definition of Neutrosophic Multiset

Let  be a universe of discourse, and .

A Neutrosophic Multiset  is a neutrosophic set where one or more elements are repeated

with the same neutrosophic components, or with different neutrosophic components.

2. Examples

is a neutrosophic set (not a neutrosophic multiset).


is a neutrosophic multiset, since the element a is repeated; we say that the element a has

neutrosophic multiplicity 2 with the same neutrosophic components.


is also a neutrosophic multiset, since the element a is repeated (it has neutrosophic

multiplicity 3), but with different neutrosophic components, since, for example, during

the time, the neutrosophic membership of an element may change.

If the element  is repeated  times keeping the same neutrosophic components ,

we say that a has multiplicity .

But if there is some change in the neutrosophic components of a, we say that a has the

neutrosophic multiplicity .

Therefore, we define in general the Neutrosophic Multiplicity Function:

where ,

and for any  one has                                     

which means that a is repeated  times with the neutrosophic components ;

a is repeated  times with the neutrosophic components , ..., a is repeated  

times with the neutrosophic components , ..., and so on.

Of course, all , and , for , with .

Also, all neutrosophic components are with respect to the set . Then, a neutrosophic

multiset A can be written as:

or .

3. Examples of operations with neutrosophic multisets.

Let's have:


1.3.1. Intersection of Neutrosophic Multisets.

1.3.2. Union of Neutrosophic Multisets.

1.3.3. Inclusion of Neutrosophic Multisets.

, but

4. Cardinality of Neutrosophic Multisets.

, and 3, where  means cardinal.

5. Cartesian Product of Neutrosophic Multisets.

6. Difference of Neutrosophic Multisets.

7. Sum of Neutrosophic Multisets.

Let's compute the neutrosophic multiplicity function, with respect to several of the

previous neutrosophic multisets.



[1] Eric W. Weisstein, Multiset, MathWorld, CRC Encyclopedia of Mathematics,

Boca Raton, FL, USA.

[2] F. Smarandache, Neutrosophic Theory and Applications, Le Quy

Don Technical University, Faculty of Information technology,

Hanoi, Vietnam, 17th May 2016.

[3] F. Smarandache, Neutrosphic Multiset Applied in Physical Processes,

Actualization of the Internet of Things, a FIAP Industrial Physics Conference,

Monterey, California, Jan. 2017.

[4] F. Smarandache, Neutrosophic Perspectives: Triplets, Duplets, Multisets,

Hybrid Operators, Modal Logic, Hedge Algebras. And Applications. Pons

Editions, Bruxelles, 323 p., 2017;

    CHAPTER X: 115-123

Neutrosophic Multiset: 115-119

Neutrosophic Multiset Applied in Physical Processes: 120-121

Neutrosophic Complex Multiset: 122-123.