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# Difference Between the Synonymous Mathematical words in a Triplet

Mathematically, one can find different words that are synonymous yet they have certain difference between them. In this blog we will analyse the triplets which are synonymous, but have a difference in the mathematical terms.

Once learners understands the difference between them, then it can be used properly and it will increase the clarity of mathematical writing.

#### Here a set of five triplets are discussed:

A. (Length, Distance, Norm): All these words have the same meaning that the shortest distance between two points A and B. Yet they must be used very carefully.

• The word ‘Length’, it is the distance between two points A and B on a real line. In other words this word stands for the length of the interval (A, B) or [A,B].

For example: Length of the interval (2, 5) or [2, 5] is =5-2=3

• The word ‘Distance’ is the Euclidean distance (shortest distance) between two points A(x 1 ,y 1 ) and B(x 2 ,y 2 ) in two dimensional space.

For Example: Distance between two points (0,1) and (3,5) is                  [(3-0)2+(5-1)2](1/2) = (9+16)(1/2) = 5

• The word ‘Norm’ is the Euclidean distance between (shortest distance) between two points A(x 1 ,x 2 ,…,x n ) and B(y 1 ,y 2 ,….,y n ) for n>2.

For Example: The norm of a vector (2,5,9,1) is its distance from origin (0,0,0,0) and is
[(2-0)
2+(5-0)2+(9-0)2+(1-0)2](1/4) = (4+25+81+1)(1/4) =3.2459

Thus these words have common meaning as a distance but they must be used depending on the underlying space, R or R 2 or R n .

B. (Minimum, Inferior, Infimum): All these words are used in set theoretic context and they carry the same meaning that minimum of a given set.

• The word ‘Minimum’ means minimum or the least value in a given countably finite set.

For Example: Min {2, 5, 8, 2.6, 4.89, 1, 2, 6, 0, 1} = 0 Set is countably finite

• The word ‘Inferior’ means minimum or the least value in a given countably infinite set.

For Example: Inferior  A = Inferior {1,1/2,1/3,1/4,……} = Lim of 1/n as n tends to infinity = 0. Here set A is countably finite.

• The word ‘Infimum’ means minimum or the least value in a given uncountable set.

For Example: Infimum A = Inf (0,1) = 0 as the set A is uncountable.

Thus these words have common meaning as the minimum but they must be used depending on the underlying set, countably finite, countably infinite or uncountable.

C. (Maximum, Superior, Supremum): All these words are also used in set theoretic context and they carry the same meaning that maximum of a given set.

• The word ‘Maximum’ means maximum or the largest (highest possible value in a given countably finite set.
• The word ‘Superior’ means maximum or the largest (highest) possible value in a given countably infinite set.
• The word ‘Supremum’ means maximum or the largest (highest) possible value in a given uncountable set.

One can work with similar examples as shown in point #B, and can get an idea about Maximum, Superior & Supremum.

Thus these words also have common meaning as the maximum but they must be used depending on the underlying set, countably finite, countably infinite or uncountable.

D. (Arrangement, Rearrangement, Derangement): These words are used in the context of number theory and combinatoric studies. Suppose two sets of entities A={a 1 ,a 2 ,…,a n } and B={b 1 ,b 2 ,….,b n } are given and problem is to establish the pair between the items in entity A and B. The above words carry important meaning.

• The word ‘Arrangement’ means the perfectly correct pairing between the items of entity A and B. It means there is no mistake in pairing. One can say that complete paring is observed in arrangement.
• The word ‘Rearrangement’ means the some of the items in entity A have perfect match with items in B and the remaining items have incorrect pairing. One can say partially pairing is observed in rearrangement
• The word ‘Derangement’ means there is not even single correct match. All matches are incorrect. One can say that in derangement zero matching is observed.

For Example: Suppose there are four letters and four envelops with names along with addresses. If someone put these letters in envelops randomly then there are possibility of all the three.
Letters = {a, b, c, d} and Envelops = {A, B, C, D}
The possibility {(a, A), (b, B), (c, C), (d, D)} is an arrangement.
The possibility {(a, A), (b, C), (c, D), (d, B)} is the rearrangement.
The possibility {(a, D), (b, C), (c, A), (d, B)} is the derangement.

Thus these words also have common base of permutations. Number of correct permutations declares the word among the three. Zero correct permutations means derangement, number of correct permutations less than ‘n’ means rearrangement and if number of correct permutations are ‘n’ in number then it is called as arrangement.

E. (Repetitions, Iterations, Recursion): These words are process oriented and mainly used in numerical methods and programming.

• The word ‘Repetitions’ refers to the mere performing same course of operations/computations again and again. Here the repetitiveness does not have any target achieving goal. In some statistical experiments this word may be refereed as ‘replications’ with a limited expectation to average out the random error.

For Example: Given the data X1, X2….., Xn computing sum of the given set of data is just the repetition of the arithmetic operation +.

• The word ‘iteration’ is mainly refereed in numerical methods. It is repetitions such that every next performance of operation/computation will come up with better approximation. It is usually used in finding the roots using Bisection or Secants method, Regula-Falsi method, Newton-Raphson method. As long as method is convergent, every iteration will give the value better close to true value as compared to earlier iterations.

For Example: Suppose the problem is obtain the solution of the equation in x3+2x-1 = 0 in the interval (0, 1).
Here f(x) = x
3+2x-1 then f ’(x) = 3x2+2 (Derivative of f(x))

On applying Newton-Raphson Method we have the following table showing iterations.

Here Newton-Raphson method iterative formula is
New (X) =Initial (X) – f(x)/f ‘(x)
From it is clear that at every iteration the value of X is coming close to the true value which is 0.453 correct up to three decimal places.

• The word ‘recursion’ is generally used in programming context. This concept involves repetition of the higher order operation/computation in terms its lower order. For example higher order factorial is expressed in terms of factorial of lower order. n! = n*(n-1)!.

Thus these words are indicative of its performance and operational practice.

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December 26, 2019

1. Very nice