Compiler Design. Computer Organization. Ethical Hacking. Computer Graphics. Software Engineering. Web Technology. Cyber Security. C Programming. Control System. Data Mining. We concentrate on the binary case. Note that you can think of all functions as relations where the input is related to the output , but not vice versa since a single element can be related to many others.
Note: symmetry and antisymmetry are not mutually exclusive: a relation can have one, both, or neither. The "divides" relation is also transitive.
Remember the relations are like functions, so it makes sense to talk about their composition, too. In the arrow diagram, every arrow between two values a and b , and b and c , has an arrow going straight from a to c. A relation satisfies trichotomy if we observe that for all values a and b it holds true that: a R b or b R a.
Given the above information, determine which relations are reflexive, transitive, symmetric, or antisymmetric on the following - there may be more than one characteristic. Answers follow. The relations we will deal with are very important in discrete mathematics, and are known as equivalence relations. They essentially assert some kind of equality notion, or equivalence , hence the name.
For a relation R to be an equivalence relation , it must have the following properties, viz. R must be:. We then proceed to prove each property above in turn Often, the proof of transitivity is the hardest. It is true that when we are dealing with relations, we may find that many values are related to one fixed value. For example, when we look at the quality of congruence , which is that given some number a , a number congruent to a is one that has the same remainder or modulus when divided by some number n , as a , which we write.
We will look into congruences in further detail later, but a simple examination or understanding of this idea will be interesting in its application to equivalence relations. We can show that congruence is an equivalence relation This is left as an exercise, below Hint use the equivalent form of congruence as described above.
However, what is more interesting is that we can group all numbers that are equivalent to each other. Above, we have partitioned Z into equivalence classes [0] and [1], under the relation of congruence modulo 2. Given the above, answer the following questions on equivalence relations Answers follow to even numbered questions. Are these special kinds of relations too, like equivalence relations?
Yes, in fact, these relations are specific examples of another special kind of relation which we will describe in this section: the partial order. For a relation R to be a partial order, it must have the following three properties, viz R must be:.
A partial order imparts some kind of "ordering" amongst elements of a set. There is some specific terminology that will help us understand and visualize the partial orders. Linear Algebra. Math Practice Questions. Table of Contents. Save Article. Improve Article. Like Article. Previous Mathematics Algebraic Structure. Next Discrete Mathematics Representing Relations.
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