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Proving Reflexive, Symmetric, and Transitive Relations in Mathematics

This guide provides step-by-step proofs for reflexive, symmetric, and transitive relations in mathematics, with examples and explanations to help understand these fundamental concepts in relation theory.

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Proving Reflexive, Symmetric, and Transitive Relations in Mathematics

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  1. Reflexive -- First sentence of proof is: • Let x Z • Let (x,x)R. • Let (x,x) I • Let x R

  2. Symmetric -- First sentence of proof is: • Let x Z • Assume (x,y)R. • Assume (y,x) R-1. • Let x R • Let (x,y) R and (y,x)  R

  3. Transitive -- First sentence of proof is: • Let x Z • Assume (x,y)R. • Assume (y,x) R-1. • Assume (x,y) R and (x,z) R • Assume (x,y) R and (y,z)  R • Assume (x,y) R and (y,x) R

  4. Reflexive -- First sentence of proof is: • Let (x,x) R2 • Let (x,y)R. • Let (x,x) R • Let x R • Let x R2 • Let (x,y)  R2

  5. Next sentence of proof is: • Assume R is symmetric. • Assume R = R-1. • Assume (x,y) R. • Assume x R. • Assume x A. • Assume (x,y)  R-1.

  6. Next sentence of proof is: • Assume R is symmetric. • Assume R = R-1. • Assume (x,y) R. • Assume x R. • Assume x A. • Assume (x,y)  R-1.

  7. Next sentence of proof is: • Assume R is symmetric. • Assume R = R-1. • Assume (x,y) R. • Assume x R. • Assume x A. • Assume (x,y)  R-1.

  8. Next sentence of proof is: • Assume R is symmetric. • Assume R = R-1. • Assume (x,y) R. • Assume x R. • Assume x A. • Assume (x,y)  R-1.

  9. Next sentence of proof is: • Assume R is symmetric. • Assume R = R-1. • Assume (x,y) R. • Assume x R. • Assume x A. • Assume (x,y)  R-1.

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