i. In business operations, set theory can assist in planning and operations. Business elements can be grouped into at least one set such as marketing, accounting, customer service, operations and within those sets, other sets. Some of these sets intersect, (e.g. sales can interest with customer service and operations sets). For example, to improve the costs of goods sold, examining where inventory intersects both sales and warehouse operations may provide the solution.
ii. Rough set theory, which postulates that all things have something in common with at least one other thing, can be used to compare and draw conclusions regarding operational elements that may not be full defined or quantified. For example, in banks, it can be used to form an early warning system of potential bank failures, thereby improving financial management.

A measure of central tendency (measure of center or central location) is a summary measure that attempts to describe a whole set of data with a single value that represents the middle /centre distribution.
The main measures of central tendency are: The mode, the median, and the mean.
i. The mode: This refers to the commonly occurring value in a distribution. For example, the mode in the following distribution (54,54,55,56,57,57,57,57,60,60) is 57, which occurs 4 times. The main advantage of mode over the median and the mean is that it can be found for both numerical and categorical (non-numerical) data.
Limitation of mode
• In some distributions, the mode may not reflect the centre of the distribution very well -as the mode may be higher or lower than the centre of the distribution when the data is ordered from lowest to the highest.
• There can be more than one mode for the same distribution data (bi-modial or multi-modia), thereby limiting the ability of the mode in describing the centre of the distribution as the centre cannot be identified.
• In some cases, particularly where the data are continuous, the distribution may have no mode at all (i.e. if all values are different).
ii. The median: The median refers to the middle value in distribution when the values are arranged in ascending or descending order. The median divides the distribution in half. For example, the median in this distribution date (54,54,55,56,57,57,57,57,60,60) is 57. The main advantage of the median is that it is less affected by outliers and skewed data than the mean and is therefore the preferred measure of central tendency when the distribution is asymmetrical.
Limitation of median
• The median cannot be identifying for categorical nominal data as it cannot be logically ordered.
iii. The mean: The mean (arithmetic mean) refers to the sum of the value of each observation in a dataset divided by the number of observations. For example, the mean of the following distribution date (54,54,55,56,57,57,57,57,60,60) =567/10 =56.7.
The main advantage of mean is that it can be used for both continuous and discrete numeric data.
Limitations of mean
• The mean cannot be calculated for categorical data, as the values can be summed
• The mean is influenced by outliers and skewed distributions as it includes every value in the distribution.

a. Decision making under certainty: This occurs when a decision maker has almost complete information and knows all the facts about the state of nature, as well as the consequences of the state of nature. Here, the problem of decision making is simple as the decision maker has only to choose the strategy which will give maximum results.
Under this condition, there is one pay-off (i.e. degree of achievements of the objective) for each strategy. The largest pay-off is chosen and the corresponding strategy is selected.
b. Decision making under risk: This arises when the state of nature is unknown. The decision makes assigns probabilities of various states of nature based on the objective or empirical evidence (i.e. historical data & past experience).
Under this condition, the strategy which gives maximum pay-off is selected.
c. Decision making under uncertainty: This arises when there is hardly any knowledge about the states of nature and no objective information about their probabilities of occurrence. Due to the absence of historical data and relative frequency, the probability of the occurrence of the particular state of nature cannot be indicated. This may occur when a new producer is introduced into the market or a new plant is set up. The decision make can only rely on market surveys and other gathered relevant information to make decisions.
Under this condition, since nothing is known about the likelihood of each state of nature, the problem becomes more complex and the personality of the decision maker plays an important role in the selection of the strategy.
d. Decision making under partial information: This arises under condition of risk and conditions of uncertainty. There is partial availability of data and the decision maker makes decisions on the basis of partial information
e. Decision making under conflict: This arises when dealing with rational opponent rather than state of nature. The decision maker therefore has to take into consideration the action or counteraction of his/her opponent when choosing a strategy (e.g. brand competition). The strategy choice is done as the basis of game theory where a decision maker anticipates the action of the opponent and then determines his/her own strategy.

i. Based on elements: These types of functions are classified based on the number of relationships between the elements in the domain and the codomain. Examples include: One-one function, Many-one function, Onto function, One-one and Onto function, Into function, and Constant functions.
ii. Based on Equations: These types of functions are classified based on algebraic  degree of the polynomial. Examples include: Identity, Linear, Quadratic, Cubic, and Polynomial functions.
iii. Based on range: These types of functions are classified based on the range which is obtained from the given functions. Examples include: Modulus, Rational, Signum, Even & Odd, Periodic, Greatest & Smallest Integer, Inverse, and Composite functions.
iv. Based on domain: These types of functions are classified based on the types of equations used to define the functions .Examples include: Algebraic, Trigonometric, Logarithmic, and Exponential functions.

i. Commerce: Matrix Cramer’s rule (determinant rule) is used in resolving various problems in economies and business involving profit maximization and loss minimization. The determinant rule is used to find solutions to linear equations (e.g. the IS-LM model used to find the combinations of interest rates and output (GDP) such that the money market is in equilibrium).
ii. Social science: Matrices can be used to record a family’s total monthly spending on different items (e.g. spending on food, health, education can be indexed by 1,2,3).
iii. Mathematics: In mathematics, matrices are used in solving linear equations and other branches of science and engineering.
iv. Science: In science, matrices are used optics to account for reflection and for reflection, electrical circuits, quantum mechanics, and resister conversion of electrical energy, etc.
v. Wireless communication: Matrices are useful to model wireless signals and to optimize them. They are used in extractions and processing of the information embedded in signals and play a key role in sensor array signal processing and the design of adaptive filters. Matrices also help in processing and representing digital images.
vi. Cryptography: Matrices are used to encrypt video signals so that only those who have video cyphers can encrypt the signals, thereby protecting satellite owners from data loss. This is done through a filtering technique which depends on matrix multiplication.
vii. Computer graphics: Matrices are used to manipulate a point in a common mathematical approach in video game graphics, as well as in expressing graphics.

i. Maximax (optimistic): This criterion is used to find the alternative that maximizes the maximum payoff. It locates the alternative with the highest possible gain and picks it.
ii. Maximin (Pessimistic): This criterion is used to find the alternative that maximizes the minimum payoff or consequence of every alternative. It first locates the payoff for each alternative and then picks the alternative with the maximum payoff. It gives the best of the worst (minimum) payoffs, guaranteeing that the payoff will be at least the maximum value.
iii. Hurwicz (Criterion of realism): This criterion is a compromise between an optimistic and a pessimistic decision coefficient of realism ( α ) is used to measure the degree of optimism of the decision maker. The weighted average is computed as follows:
Weighted average = (α) x (maximum in row) + (1-α) x (minimum in row)

iv. Laplace (Equally likely): This criterion uses all the layoffs for each alternative and assumes that all probability of occurrence for the state of natures are equal (i.e. equally likely). It finds the average payoff for each alternative and selects the alternative with the highest average.
v. Minimax regret: This criterion is bases on opportunity loss or regret (i.e. the amount lost by not picking the best alternative in a given state of nature).it fist creates the opportunity loss table, then finds that alternative that minimizes the maximum opportunity loss within each alternative.

b. Number of students who played at least two of the sports
=15 +6+60+18
=99
c. Number of students who played one sport only
=303 +126 + 195
=624
d. Number of students who did not play any of the three sports.
=840-(624+99)
=117