Topic 2 : Look into equivalence relations, partitions, equivalence classes and related theorems
Equivalence relations
Definition
In mathematics, an equivalence relation is a binary relation between two elements of a set in which groups them together as being "equivalent" in some way. Let a, b, and c be arbitrary elements of some set X. Then "a ~ b" or "a ≡ b" implies that a is equivalent to b.[1]
An equivalence relation on a set is a subset of , i.e., a collection of ordered pairs of elements of , satisfying certain properties. Write " " to denotes is an element of , and this is called " is related to ," then the properties of this thing are
Reflexive: for all ,
In mathematics, a binary relation R over a set X is reflexive
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A reflexive relation is said to have the reflexive property or is said to possess reflexivity.[3]
Symmetric: implies for all
In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that a is related to b if and only if b is related to a.
In mathematical notation, this is:
Ɐa,b ϵ X(aRb ↔ bRa) [4]
Transitive: and brings about for all ,
In mathematics, a binary relation R over a set X is transitive if an element a is related to an element b, and b is in turn related to an element c, then a is also related to c. Transitivity is a key property of both partial order relations and equivalence relations.
In terms of set theory, the transitive relation can be defined as:
Ɐa,b,c ϵ X: (aRb ˄ bRc) → aRc [5] where these three properties are completely independent. Other notations are often used to indicate a relation, e.g., or .[2]
EXAMPLE
Let the set {a, b, c} have the equivalence relation {(a,a), (b,b), (c,c), (b,c), (c,b)} . The following sets are equivalence classes of this
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We also have Bn=∑_(k=1)^n▒〖S (n,k)〗 Set partitions can be ordered, that is to say ({1,2},{3})≠({3},{1,2}), and the number of ordered set partitions is ∑_(k=1)^n▒〖k!S(n,k)〗 A set partition is said to be non-crossing if the parts can not be interlaced. That is to say there does not exist P,Q∈Pwhich contains elements ,b∈P and x,y∈Q such that a that the equivalence classes form a partition of X. This partition – the set of equivalence classes – is sometimes called the quotient set or the quotient space of X by ~ and is denoted by X / ~.
When the set S has some structure (such as a group operation or a topology) and the equivalence relation ~ is defined in a manner suitably compatible with this structure, then the quotient set often inherits a similar structure from its parent set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.[8]
Example
Consider the set
S={0,1,2,3,4,5}.
There are many equivalence relations we could define on this set. One would be xRy⇔x=yxRy⇔x=y, in which case the equivalence classes
...r and they willing to help her when she is in trouble. For example, she willing to satisfice her time to celebrate her friend’s birthday until late of the night. According to Papalia (2009), this relationship known as fictive kin. Fictive kin used to describe individuals that is not related by birth or marital statuses but have a significant emotional connection with that other individual and it would develop the characteristics of a family relationship.
The definition of truth is the epitome of what we we all perceive to be reality. Truth is what we sometimes think about in the back of our head, but we are unsure of whether this truth is really “true.” The objective correlative is another term that is used to refer to truth. The objective correlative is getting enough people to feel the same emotion and agree upon it. Objective correlative also refers to the objective truth or the facts. It’s trying to get the subjective truth to become objective, then subjective for each person. The idea of objectivity is that it is concrete, measurable, and tested. This idea of objectivity relates to the characteristics of what facts are.
Relations of ideas are parts of knowledge that are a priori, or not learned by experience. “Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe.”(500) So geometry would be put in this category because of the relationship between lines and figures. The opposite this type of knowledge is logically impossible. The example of how a square can never be a circle shows how the opposite of these claims do...
Symbiosis is a term that means “living together”. Symbiotic relationships can be mutualistic (both species benefit), commensalism (one species benefits, the other neither benefits nor suffers to any extent) or parasitic (one species associates with another causing harm to the host organism).
o The terms of the classification tell us what the individuals in that class have in common.
The term ‘analogy’ is very vague in nature, but when used in this context, we assume that the behavior of other people is in many ways analogous in reference to causes. These causes being behavior directed from sensation or thought. It is apparent and observable that people or beings other then I behave in ways in which we behave when placed in different situations. For example sadness or the nature of anger or happiness can be seen in others. Others then can and do react to different causes similar to the way in which I do as well. Another consideration is that of shared experience. Russell uses the example of two friends having a conversation in which memoirs are explored. These two individuals have shared experiences together. They eventually discover that each other’s memories aid each other in recalling information forgotten with time. (Russell 89)
Sibling relationships relates to brothers sisters and in a lot of families in this era step-brothers and step-sisters. Family is hopefully the other members of each side of the parent’s family who are involved with the child at some point, in some families this is a very strong relationship due to religion or beliefs and in others distance or other reasons may mean that family are not around as much. Friendships, these hopefully are being built even as babies if babies are taken to groups to socialise and learn how to be with other babies and children. Emotional relationships are the ones that affect the child the most so if a parent shouts for one reason or another the child will feel upset as they do not like a person they care about raising their voice. Acquaintances can be anyone from the health visitor to the post person if they are a regular person who is seen every so often, my son has to see the bin men on a Friday morning and watch them empty our bins and say thank you.
A common relationship I think of when I think of relationships is friendship. I have many friends that have different traits and characters which forces me to have different relationships with each and every one of them. Having a relationship with your friends may be a negative or positive relationship. For example, if you have a disagreement with one of your friends, you may have a negative vibe towards them, which would create a negative relationship with them. If there is a positive relationship, usually both people in the “relationship” have common likes/dislikes and can agree on one
Have you ever believed in something so strongly that when someone would say something different, it would not influence your decision? If so, you may have experienced the first feature of reality, Reflexivity, in which one has an incorrigible proposition. Reflexivity is a relation that exists between entity and itself. For exa...
similarities in which they can relate, but not all cultures are the same. There are many factors that can
Law of Proximity: this law states that items which are close together in space or time tend to be perceived as belonging together or forming an organized group.
distinguishes a tie between people that will provide that interaction as well as keep the interaction consistent on a regular basis.
Intersubjectivity: Intersubjectivity is also an aspect of social interaction whereby two people who are taking part in an individual task come to a shared comprehension. The two people at the beginning of the task may have a different understanding but eventually, arrive at a common one (Berk, 2013).
By definition, symbiosis is the close and often long term interaction between two or more different species. The word symbiosis is derived from the Greek words, sym (together) and bio (life). German biologist Anton de Bary is the scientist typically credited with coining the term in the middle of the nineteenth century, though also an important contribution was made by his contemporary, Albert Bernhard Frank. Frank, however, instead used the term symbiotism. De Bary introduced the term to discuss lichens, which he discovered were parts of algae and fungi. On the contrary, Frank reported on the association between fungi and roots of forest trees, which he named mykorrhiza. Mycorrhiza is also a Greek word combined from the two words meaning fungus and roots. The association in mycorrhiza is a significant component of soil life and soil chemistry, and the plant roots characterized by it are from vascular plants.
Words that have to do with logic like and, or, not are given symbols like &, V, or an upside down L reversed. The Letters X, Y, Z and so on are commonly used as variables and P, Q, R are used as predicates, properties or relations.