| Summary: | Many real-life problems are well represented only by sets which allow repetition(s), such as the multiset. Although not limited to the following, such cases may arise in a database query, chemical structures and computer programming. The set of roots of a polynomial, say <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, has been found to correspond to a multiset, say <i>F</i>. If <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>g</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> are polynomials whose sets of roots respectively correspond to the multisets <inline-formula> <math display="inline"> <semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, the set of roots of their product, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mi>g</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, corresponds to the multiset <inline-formula> <math display="inline"> <semantics> <mrow> <mi>F</mi> <mo>⊎</mo> <mi>G</mi> </mrow> </semantics> </math> </inline-formula>, which is the sum of multisets <i>F</i> and <i>G</i>. In this paper, some properties of the algebraic sum of multisets ⊎ and some results on selection are established. Also, the count function of the image of any function on Dedekind multisets is defined and some of its properties are established. Some applications of these multisets are also given.
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