When we think of cricket we start thinking about objects like a bat, ball, stumps, teams, stadium, etc. But like cricket, we play football on the ground and like cricket in teams also. Then what makes the game of football different from cricket.
Obviously, each game is played with specific equipment and with specified rules separately defined for each game. And that’s how the term of sets is used in mathematics. It’s a collection of well-defined objects. Like in the above case we have a set of cricket related things. The set can only be called as a set if its all member follow the same rule. Before establishing any set, one must define the criteria by which all the entities will be grouped together.
There is a number of examples of sets can be given that we actually exercise in our day to day life. Be it a grocery list or the set of all the chicken recipes or musical hits by Michael Jackson or a set of TV shows. Further, there are shows watched by elders and the shows most watched by children. The subcategorized sets in mathematics are called as subsets. As done in mathematics, we can apply mathematical operations to any set once the sets are defined. The mathematical operations include union, the intersection of sets, etc.
In schools, biological texts hardly mention the systematics – the study of all the living organisms is an application of biomath as set theory for the classification of the living organisms on earth. Be it a five Kingdom system proposed by R.H. Whittaker or a three-domain system introduced by Carl Woese.
Nucleic acid molecules like DNA and RNA are polymers of nucleotides. The nucleotides include A- adenine, T- thymine, G- guanine, C- cytosine, and U- uracil. Depending upon their chemical structures they’re a part of either DNA or RNA. If we define a set of all the nucleotide it will be represented as
N={ATGCU}
Here, we can define other sets as well,
P={AG}
D={TCU}
Set P is a set of the Purines and D is a set of pyrimidines present in nucleic acids.
Once the sets are defined mathematical operations can be performed on sets. For example, the union of sets N and P would be as follows.
N U P = {ATGCU},where U means union of sets N and P.
Also the union of set N and D would be,
N U D = {ATGCU}.
Here the outcome of the operation of the union of set N with set P or D is set N, which means that set P or D are part of set N. meaning they are the subset of set N.
N U P = N or N U D = N.
The subsets are represented as P ⊆ N or D ⊆ N.
Also, we can apply the operation of the subtraction to this example. Which is represented as,
N – P = {TCU}
And subtraction of set N and D would be,
N – D = P={AG}.
From the above example, we can also make out that set P and D both don’t have any common element (a member of a set). But the union of both the sets gives us,
P U D= {ATGCU} = N.
This means that set P and D are complementary to each other. Set P is a compliment of set D with respect to N. Compliment P is represented as P’ and complement of D is represented as D’.
In the laboratory, we all have performed the experiment and prepared a standard graph required to determine the concentration of DNA or protein from test samples. Here we have two different sets. One set comprises the values of different concentrations of the reference standard. The second set that we obtain is a set of OD or absorbance values by performing an experiment.
Here we establish a particular relationship between the values of two sets that is concentrations of the reference standard and OD or absorbance. And this relationship is now used to find the concentration of the corresponding analyte in the test sample.
In the mathematical language, this specific relationship between two sets is termed as FUNCTION. Where one input value from one set has only one output value from the other set. There are different functions used as a part of science math studies. In the next article, we will discuss what mathematical functions mean in biology in detail, and how they are useful in biological studies.
Here detailed math and science procedures are not of our interest. The only purpose of this article is to show that sets are formed from biological data proving that mathematics in science will complement the research aptitude. All operational procedures applied to the data set can be found in any mathematical textbook but we cannot treat it as plane mathematics only.
Further reading:
Set Theory, Logic, and Probability: The Integration of Qualitative Reasoning into Teaching Statistics for Quantitative Biology. CBE Life Sci Educ December 1, 2016 15:le3, DOI:10.1187/cbe.16-06-018
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5132388/pdf/le3.pdf
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