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Expert: - 1/9/2008


Question

graph relations
I have a bit of trouble understanding what the question is asking, exactly? What type of relation is each graph, and why?

Thanks! :)

Answer
Questioner:   Sam A
Category:  Advanced Math
Private:  No
 
Subject:  Graph Relations
Question:  

graph relationsI have a bit of trouble understanding what the question is asking, exactly? What type of relation is each graph, and why?

Thanks! :)  
............................................................
Hi, Sam,

I think that you are asked to test each relation, R, to see if it passes the tests for:

A. Reflexive: Always: x R x
B. Symmetric: Always:  if x R y then y R x
C. Transitive: Always: if x R y and y R z then  x R z.
D. Antisymmetric: Always: if a R b and b R a, then a = b (i.e. a, b are the same object)



And in the picture for each relation,  an arrow from x to y means x R y.

(i) shows  a R b and b R a, so it is symmetric.  NOt ref or trans, not antisym.

(ii) a R b, b R c, c R a, and d R d.  Not sym or ref, but trans.  
Note: d R d and d R d -> d R d is true.  it is antisym, because we never have a R b and b R a, except for d R d.

(iii) a R a, a R b, b R b.  Reflexive, not sym.  Antisym. a R b, but not b R a.

(iv) a R a, b R b, b R c, c R b, c R c  Ref, sym, and trans. Not antisym.


For partial ordering, see if we have ref, antisym, and trans.  I think only (iii) qualifies.

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Paul Klarreich

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I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

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