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A detailed tutorial on the universe of discourse. Step by step tutorial including several examples of the universe of discourse for reference.

The universe of discourse is normally just referred to simply as the universe of a set. The universe of discourse can also be the universe of a certain truth set. Basically, it is all possible considerations for a truth set. It is also a set itself, one where many different subsets are taken from. As you can tell, the universe of discourse has different meanings depending on the exact branch of math you are studying. However, all definitions have one thing in common: the universe is a set where many other sets are taken from. Normally it is easy to figure out what the universe of dicourse is based on the context of the problem you are trying to solve.

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A detailed tutorial on ordinary sets in set theory. Step by step tutorial including several examples of ordinary sets in set theory for reference.

You may be reading this and asking yourself, what is an ordinary set? An ordinary set is a set where the complete set is not part of the set. This is not the same as a subset, for as we know all sets are subsets of themselves. An example of an ordinary set is the set of all pencils. The set of pencils is not a pencil, so it is considered an ordinary set. However, the set of all thoughts is a thought. So, that set is not ordinary. In general, all sets are ordinary sets except for certain thoughts and concepts.

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A detailed tutorial on the trichotomy property. Step by step tutorial including several examples of the trichotomy property for reference.

The trichotomy property is one of the ordering properties of natural numbers. It tells us what order you need to put the natural numbers in – in other words, it tells you the placement of each element of the set of natural numbers. The trichotomy property states that is there are two natural numbers m and n, that m must be either less than n, equal to n, or greater than n. The smaller number is to be placed first, with the larger number after it. If the numbers are equal, then only one number needs to be included as part of the set.

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A detailed tutorial on how to construct a Cayley table. Step by step tutorial including several examples of how to construct a Cayley table for reference.

A Cayley table is a table that expresses the structure of a finite set. A Cayley table is set up by having the elements of the set across the first row, and numbers going in a numerical order of n + 1 starting at 1 down the first column. Sometimes the table is simply different ways the elements can be ordered. Other times is is a true table, where an operation is performed between two numbers in the space where they cross each other. However, a true Cayley table must be constructed using an identity skeleton. Once an identity skeleton for the finite set has been decided on, the Cayley table can be filled out using the identity skeleton. Since there is more than one possible identity skeleton for a finite set, you may have to go through a trial and error process until you find the right one.

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A detailed tutorial on the definition of cardinal numbers. Step by step tutorial including several examples of how to define cardinal numbers for reference.

Cardinal numbers are natural numbers that are used to measure cardinality of sets. Cardinality is a fancy way of saying the size of a set. This means the cardinality is the number of elements in a set, provided that the set is finite. If the set is infinite, something called a transfinite cardinal number is used to describe the cardinality of the set. Cardinal numbers are a very important part of set theory, even though they are not studied often or used constantly.

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A detailed tutorial on the mathematical study of topology. Step by step tutorial including several examples of topology for reference.

Topology is a study in mathematics that deals with space and spatial properties of objects. There are several different types of topology. The most common topics, called subtopics, are point-set topology, algebraic topology, geometric topology, and low dimensional topology. Topology may be a familiar sounding name to you – doubtless you have heard of a “topographical map,” used in science classes. However, the way the topographic map is created is with the study of math known as topology.

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A detailed tale of the Infinite Hotel. Step by step story including several pictures and an explanation of the Infinite Hotel for reference.

The Infinite Hotel is a famous math story and puzzle that was thought of by David Hilbert, a German mathematician. Sometimes the Infinite Hotel is called Hilbert’s Paradox of the Grand Hotel. It states that if one person comes into the hotel and all the rooms are full, they can all move down one room and the person can then take the first room. If k number of people come into the hotel and all the rooms are full, everyone can move down k number of rooms to make room for the people that just arrived. And, if double the amount of people that are already there are looking for rooms, everyone in room n can move to room 2n, making room for all the new arrivals in the odd-numbered rooms. This example of the Infinite Hotel can be used in certain forms of mathematical induction, and also in set theory and studies dealing with infinite numbers.

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A detailed tutorial on the inverse image of sets. Step by step tutorial on the inverse image of sets for reference. Knowledge of the inverse image of sets is important in advanced discrete mathematics courses.

Say that you have a function f: A –> B. Then, X is a subset of A and Y is a subset of B. The image of X or the image set of X is f(X) = {y belongs to B: y = f(x) for some x belonging to X}. The inverse image of Y is defined as f^-1(Y) = {x belongs to A: f(x) belongs to Y}. The inverse image is simply a reversed form of the image. Often when asked to find the inverse image, it will help to set up a drawing of the image of the function, connecting everything where it needs to go. Then to find the inverse you simply reverse your work.

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A detailed tutorial on the preimage of a set. Step by step tutorial including several examples of the preimage of a set for reference.

The preimage of a set is defined over a function. If there is a function over A and B, then we can say that y = f(x), provided that (x, y) belongs to f. Based on this definition, x is the preimage of y under f. To find the preimage, simply look for the value of x that matches with the proper value of y in any function of ordered pairs in A and B.

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A detailed tutorial on the definition of an outlier. Step by step tutorial including several examples of definitions of outliers for reference.

An outlier is a type of observation of statistical data. It is usually very far away from the other values in the data set, hence the name. Usually it is a number that is much smaller than the other numbers, although it could be much larger than the other numbers as well. Outliers have an equal chance of occuring in any random observation, but they are still rare. Typically when an outlier is found it means there is some sort of mistake, usually a measurement error.