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My note have define the binary relation as follow but I find out I cannot understand the meaning of the symbols.

let R be a binary relation on A

what is a binary relation? My note only said "A relation R on A x B is called a binary relation form a to B. If A = B then we simply say R is a binary relation on A". if this is only relation on A where is the x,y,z (below) come form.

R is reflexive iff (x,x) for all x in A $\forall x [(x,x) \in R]$ is true

what is (x,x) mean? and what is "is true" mean?

R is symmetric iff $\forall x \forall y [(x, y) \in R \Rightarrow (y,x) \in R]$

what is the different between (x,y) and (y,x)? and why using$ \Rightarrow$ insteat of $\rightarrow$?

R is antisymmetric iff $\forall x \forall y [((x,y)\in R \land (y,x)\in R)\Rightarrow (x=y)]$

what is antisymmetric mean? Why is the defination so different form the symmetric?

R is transitive iff $\forall x \forall y \forall z[((x,y)\in R \land (y,z)\in R)\Rightarrow (x,z)\in R]$

why the result is only (x,z)? where is the y?

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Have you tried contacting your professor or TA? –  Asaf Karagila Oct 18 '12 at 14:23
    
I have try seem they are not willing answer. May be my question is so fundamental, they thought the student should know. Sad. –  Samuel Oct 18 '12 at 14:54
    
Any relevant elementary text book will answer these questions. You should always read a good text book or two alongside your lecture notes: seeing more than one presentation of basic ideas will much improve your understanding of what's going on, whatever the topic in maths. –  Peter Smith Oct 18 '12 at 15:24

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A binary relation on $A$ is a subset of $A\times A$. As such it is a set whose elements are pairs. If $a_1,a_2\in A$ then $(a_1,a_2)$ is a pair of elements of $A$, hence an element of $A\times A$; therefore it makes sense to ask whether or not $(a_1,a_2)$ is an element of that subset $R$ of $A\times A$.

Thus $(x,x)$ with $x\in A$ is a special case of such pairs, namel ya pair wher both components are the same. If all such pairs are in $R$, then $R$ is called reflexive. In formal symbolism, this is written as $\forall x\in A\colon (x,x)\in R$. Thus $R$ is reflexive iff $\forall x\in A\colon (x,x)\in R$ is true.

We are concerned with ordered pairs here, that is a pair whose first componement is $x$ and second component is $y$ is something different than a pair whose first componement is $y$ and second component is $x$ (at least if $x\ne y$).

Some authors make a strict distinction between $\Rightarrow$ and $\rightarrow$ (cf. for example some discussion at Wikipedia), some don't. From your level of question I consider it save to assume that no distinction is necessary. Thus in short: Both symbols are used to denote implication ("if ... then").

The meaning of antisymmetric is precisely given by the formal definition stated. Once you will have inhaled these definitions a bit, the difference in the surface between the definitions will go away. Symmetric means that $(x,y)$ and $(y,x)$ are always both or neither in $R$; antisymmetric means that they are never both in $R$ (with the exception of pairs with $x=y$).

In the definition of transitivity, there is just no room for $y$ on the right side. Remember that $R$ contains pairs, not tripes of elements of $A. Here again, digesting a few examples smight make things clearer. If "Jack is taller than Jill" and "Jill is taller than Joe", you can conclude that "Jack is taller than Joe" and there is no need to mention Jill in that result.

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Thanks, start to understand my note. ^^ –  Samuel Oct 18 '12 at 16:15

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