GO Classes Set Theory And Algebra Practice Set 1 | Question: 24

Question

For each of the following relations $R$ on the set of real numbers, decide whether it is reflexive, symmetric, and/or transitive? Justify your arguments. Is the relation an equivalence relation? Explain.

• A. $(x, y) \in R$ if and only if $|x-y| \leq 3$.
• B. $(x, y) \in R$ if and only if $x \cdot y>0$.
• C. $(x, y) \in R$ if and only if $x^2-y=y^2-x$.
• D. $(x, y) \in R$ if and only if $(x-y)\left(x^2+y^2-1\right)=0$
• E. $(x, y) \in R$ if and only if $|x+y|=|x|+|y|$.

Answer 1

• A. Reflexive, symmetric but not transitive 
• B. Not reflexive but symmetric and transitive
• C. Reflexive, symmetric, transitive 
• D. Reflexive, symmetric, transitive
• E. Reflexive, symmetric, transitive