Let R be a relation on a non-empty collection of sets. It is defined as A R B…
1996
Let R be a relation on a non-empty collection of sets. It is defined as A R B if and only if A intersection B is the empty set. Which statement is true?
- A.
R is reflexive and transitive
- B.
R is symmetric and not transitive
- C.
R is an equivalence relation
- D.
R is not reflexive and not symmetric
Attempted by 48 students.
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Correct answer: B
for symmetry, if A R B, then A intersection B is empty. since intersection is commutative, B intersection A is also empty, so B R A. hence R is symmetric.
R is not reflexive in general. A R A would mean A intersection A is empty, but A intersection A = A. this is true only for the empty set, not for every set in the collection.
R is not transitive. for example, take A = {1}, B = {2}, and C = {1}. then A intersection B is empty and B intersection C is empty, so A R B and B R C. but A intersection C = {1}, so A R C is false.
therefore, R is symmetric and not transitive.