Prove that if R is a symmetric, transitive relation on A and the domain of R is A, then R is reflexive on A. I'm not really sure where to start.
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Originally Posted by Ryaη Prove that if R is a symmetric, transitive relation on A and the domain of R is A, then R is reflexive on A. If then by the fact that . By symmetry we know that . Now you finish.
Originally Posted by Plato If then by the fact that . By symmetry we know that . Now you finish. Would you then go on to say: Since and and since transitivity holds on R, then xRy and yRx xRx (which is the definition of being reflexive)? Therefore R is reflexive on A.
Originally Posted by Ryaη Would you then go on to say: Since and and since transitivity holds on R, then xRy and yRx xRx (which is the definition of being reflexive)? Therefore R is reflexive on A. yup i'd word it to fit the definition more, you know, with quantifiers and all that, but yes, that's the basic idea
How could I word this to fit the definition better? Let . . By symmetry we know . Since and , then by transitivity we know (xRy and yRx) xRx. (definition of being reflexive) Therefore R is reflexive on A.
Originally Posted by Ryaη How could I word this to fit the definition better? Let . . By symmetry we know . Since and , then by transitivity we know (xRy and yRx) xRx. (definition of being reflexive) Therefore R is reflexive on A. look at how the definitions are in your text, for instance, they usually end with phrases like "for all "
It's getting kind of late here. Now that I have the main idea down, I'll work on refining it tomorrow. Thank you for all the help!
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