Restricting a relation to a subset
The restriction of R to a subset B ⊆ A is R ∩ (B × B): keep only the pairs with both coordinates in B. Restriction inherits every property defined by a universally quantified condition — symmetry, antisymmetry, transitivity and irreflexivity all survive, and so does reflexivity. What can change dramatically is structure that depends on the whole set, such as which elements are maximal or whether suprema exist.
✓ Unlimited questions · marked criterion by criterion · no card needed
Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Intersect with B × BDelete every pair with a coordinate outside B. Keeping pairs that only half-belong is the usual slip.
- Argue property inheritanceA universally quantified property that holds for all of A holds for the subset, since fewer instances need checking.
- Recheck existence claimsMaximal elements, least upper bounds and connectedness are not inherited — they must be recomputed inside B.
- Watch reflexivity’s domainThe restricted relation is reflexive on B, not on A. State which set you mean.
Worked example
On A = {1,…,6} let R be divisibility. Restrict to B = {2,3,4} and find the maximal elements in each.
- Restriction keeps pairs (a,b) with a ∣ b and both in B: (2,2),(3,3),(4,4),(2,4).
- Maximal elements of B under this restriction: 3 (nothing above it) and 4.
- In the full A, 4 is not maximal — 4 ∣ 4 only, but 5 and 6 are also maximal, and 2 ∣ 6 so 2 is not.
- The maximal elements of A are 4, 5 and 6.
Answer. The restriction is {(2,2),(3,3),(4,4),(2,4)}, with maximal elements 3 and 4 — different from the maximal elements 4, 5, 6 of the full poset.
Where marks get dropped
These are the specific errors that cost credit on restricting a relation to a subset questions — QED's rubric penalises each of them separately.
- Keeping pairs with only one coordinate in B. Both must be in B for the pair to survive.
- Assuming maximal or least elements are inherited. They depend on what else is present and must be recomputed.
- Confusing restriction of the domain with restriction of both coordinates. R ∩ (B × A) restricts only the source, which is a different operation.
Practise this until it is automatic
Unlimited fresh questions
QED generates new restricting a relation to a subset problems on demand at warm-up, exam and challenge level, so you can drill this one skill until it stops costing you marks.
Marked like an examiner
Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.
Answer in real notation
A one-tap symbol palette, a visual equation editor and a truth-table builder — or photograph your handwritten working and QED converts it to LaTeX.
Saved to your library
Every question you generate is kept and re-takeable as a timed exam, and your Relations mastery is tracked so you know when this is exam-ready.
Restricting a relation to a subset — frequently asked questions
Which properties are always inherited?
Reflexivity (on B), irreflexivity, symmetry, antisymmetry and transitivity — everything expressible as a ∀-statement about pairs and triples already in the relation.
Can a restriction be a total order when the original is not?
Yes, and that is the definition of a chain. Restricting divisibility on {1,…,6} to {1,2,4} gives a total order even though divisibility is only partial.
Does restriction preserve being a lattice?
No. A sublattice must be closed under meet and join, which restriction does not guarantee — the supremum of two elements of B may lie outside B.
The rest of Relations
Properties of relations, composition, and representations. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Reflexive, symmetric, antisymmetric & transitive
- 2Checking properties for a given relation
- 3Composition R∘S & inverse R⁻¹
- 4Matrix & digraph representations
- 5Reflexive & transitive closures
- 6Relations as subsets of A × B
- 7Powers Rⁿ & reachability
- 8Warshall’s transitive closure algorithm
- 9Counting relations with a given property
- 10Union, intersection & complement of relations
- 11n-ary relations & the relational data model
- 12Restricting a relation to a subset
- 13Symmetric closure vs transitive closure
Ready to make restricting a relation to a subset exam-proof?
Generate your first questions free — no card, no setup, no personal data stored. Practise until the method is second nature.
Start practising free →