Let R be a non-empty relation on a collection of sets defined by A R B...
Explanation:
To understand the properties of the relation R on a collection of sets, let us consider the following:
Reflexive property: A relation R is said to be reflexive if (a, a) ∈ R for every a ∈ A.
Transitive property: A relation R is said to be transitive if (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R for every a, b, c ∈ A.
Symmetric property: A relation R is said to be symmetric if (a, b) ∈ R implies (b, a) ∈ R for every a, b ∈ A.
Equivalence relation: A relation R is said to be an equivalence relation if it is reflexive, symmetric and transitive.
Now, let us consider the relation R defined by A R B if and only if A ∩ B = φ.
Symmetric and not transitive:
To prove that R is not reflexive, we can consider a set A such that A ∩ A = φ, which is not possible since every set has at least one element in common with itself. Hence, (A, A) ∉ R for any set A, and R is not reflexive.
To prove that R is not symmetric, we can consider two sets A and B such that A ∩ B = φ, but B ∩ A ≠ φ (i.e., B and A have a common element). Hence, (A, B) ∈ R but (B, A) ∉ R, and R is not symmetric.
To prove that R is symmetric and not transitive, we can consider three sets A, B, and C such that A ∩ B = φ, B ∩ C = φ, but A ∩ C ≠ φ (i.e., A and C have a common element). Hence, (A, B) ∈ R and (B, C) ∈ R, but (A, C) ∉ R, and R is not transitive.
Therefore, the only option that is true is B, which states that R is symmetric and not transitive.