**The Neutrosophic Multisets and the Neutrosophic
Multiset Algebraic Structures were introduced **

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

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).

But

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.

While

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:

Then:

### 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.

.

**References**

[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, 17^{th} 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.**__
__