01 Key Concepts
Definition
A relation from set A to set B is a set of ordered pairs (a,b), where a is in A and b is in B.
Domain and Range
The domain is the set of all first elements (a-values) used in the relation; the range is the set of all second elements (b-values).
Reflexive
A relation on set A is reflexive if every element is related to itself: (a,a) is in the relation for all a in A.
Symmetric
A relation is symmetric if whenever (a,b) is in the relation, (b,a) is too.
Transitive
A relation is transitive if whenever (a,b) and (b,c) are both in the relation, (a,c) must be too.
Equivalence Relation
A relation that is reflexive, symmetric, AND transitive all at once — these relations partition a set into distinct groups.
02 Solved Examples
Example 1 Let R = {(1,1),(2,2),(3,3),(1,2),(2,1)} on set {1,2,3}. Is R reflexive?
- Check if (1,1), (2,2), and (3,3) are all present.
- All three pairs are in R.
Answer: Yes, R is reflexive
Example 2 Using the same R, is it symmetric?
- Check each pair: (1,2) is in R, and (2,1) is also in R.
- All other pairs are of the form (a,a), which are trivially symmetric.
Answer: Yes, R is symmetric
Example 3 Let R = {(1,2),(2,3)} on {1,2,3}. Is R transitive?
- Check if (1,2) and (2,3) being present requires (1,3) to be present.
- (1,3) is not in R.
Answer: No, R is not transitive
03 Practice Questions
1Find the domain of R = {(1,2),(3,4),(5,6)}.
{1,3,5}
2Find the range of R = {(1,2),(3,4),(5,6)}.
{2,4,6}
3Is R = {(1,1),(2,2)} on {1,2} reflexive?
Yes
4Is R = {(1,2)} on {1,2} symmetric?
No, since (2,1) is not in R
5What three properties define an equivalence relation?
Reflexive, symmetric, and transitive
📄 Relations — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.