1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Show that the relation R on the set A = {x ∈ Z ; 0 ≤ x ≤ 12}, given by R = {(a, b) : a = b}, is an equivalence relation. Find the set of all elements related to 1.

Open in App
Solution

We observe the following properties of R. Reflexivity: Let a be an arbitrary element of A. Then, $a\in R\phantom{\rule{0ex}{0ex}}⇒a=a\left[\mathrm{Since},\mathrm{every}\mathrm{element}\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{itself}\right]\phantom{\rule{0ex}{0ex}}⇒\left(a,a\right)\in R\mathrm{for}\mathrm{all}a\in A\phantom{\rule{0ex}{0ex}}\mathrm{So},R\mathrm{is}\mathrm{reflexive}\mathrm{on}A.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Symmetry}:\mathrm{Let}\left(a,b\right)\in R\phantom{\rule{0ex}{0ex}}⇒ab\phantom{\rule{0ex}{0ex}}⇒b=a\phantom{\rule{0ex}{0ex}}⇒\left(b,a\right)\in R\mathrm{for}\mathrm{all}a,b\in A\phantom{\rule{0ex}{0ex}}\mathrm{So},R\mathrm{is}\mathrm{symmetric}\mathrm{on}A.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Transitivity}:\mathrm{Let}\left(a,b\right)\mathrm{and}\left(b,c\right)\in R\phantom{\rule{0ex}{0ex}}⇒a=b\mathrm{and}b=c\phantom{\rule{0ex}{0ex}}⇒a=bc\phantom{\rule{0ex}{0ex}}⇒a=c\phantom{\rule{0ex}{0ex}}⇒\left(a,c\right)\in R\phantom{\rule{0ex}{0ex}}\mathrm{So},R\mathrm{is}\mathrm{transitive}\mathrm{on}A.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$ Hence, R is an equivalence relation on A. The set of all elements related to 1 is {1}.

Suggest Corrections
0
Join BYJU'S Learning Program
Related Videos
Equivalence Class
MATHEMATICS
Watch in App
Explore more
Join BYJU'S Learning Program