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Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = φ. Then, (pick the true statement) A: R is reflexive transitive B: R is symmetric and not transitive C: R is an equivalence relation D: R is not reflexive and not symmetric
Answer is option 'B'. How !?
Answer is option 'B'. How !?
Most Upvoted Answer
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.
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.
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Let R be a non-empty relation on a collection of sets defined by A R B...
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Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = φ. Then, (pick the true statement) A: R is reflexive transitive B: R is symmetric and not transitive C: R is an equivalence relation D: R is not reflexive and not symmetric Answer is option 'B'. How !?
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Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = φ. Then, (pick the true statement) A: R is reflexive transitive B: R is symmetric and not transitive C: R is an equivalence relation D: R is not reflexive and not symmetric Answer is option 'B'. How !? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = φ. Then, (pick the true statement) A: R is reflexive transitive B: R is symmetric and not transitive C: R is an equivalence relation D: R is not reflexive and not symmetric Answer is option 'B'. How !? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = φ. Then, (pick the true statement) A: R is reflexive transitive B: R is symmetric and not transitive C: R is an equivalence relation D: R is not reflexive and not symmetric Answer is option 'B'. How !?.
Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = φ. Then, (pick the true statement) A: R is reflexive transitive B: R is symmetric and not transitive C: R is an equivalence relation D: R is not reflexive and not symmetric Answer is option 'B'. How !? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = φ. Then, (pick the true statement) A: R is reflexive transitive B: R is symmetric and not transitive C: R is an equivalence relation D: R is not reflexive and not symmetric Answer is option 'B'. How !? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = φ. Then, (pick the true statement) A: R is reflexive transitive B: R is symmetric and not transitive C: R is an equivalence relation D: R is not reflexive and not symmetric Answer is option 'B'. How !?.
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Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = φ. Then, (pick the true statement) A: R is reflexive transitive B: R is symmetric and not transitive C: R is an equivalence relation D: R is not reflexive and not symmetric Answer is option 'B'. How !?, a detailed solution for Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = φ. Then, (pick the true statement) A: R is reflexive transitive B: R is symmetric and not transitive C: R is an equivalence relation D: R is not reflexive and not symmetric Answer is option 'B'. How !? has been provided alongside types of Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = φ. Then, (pick the true statement) A: R is reflexive transitive B: R is symmetric and not transitive C: R is an equivalence relation D: R is not reflexive and not symmetric Answer is option 'B'. How !? theory, EduRev gives you an
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