text: Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that A x (B ∩ C) = (A x B) ∩ (A x C)
  
  Asked by Aaryan | 1 year ago |  55
  
  ##### Solution :-
  
  A × (B ∩ C) = (A × B) ∩ (A × C)
  
  Let us consider LHS A × (B ∩ C)
  
  (B ∩ C) = ∅
  
  A × (B ∩ C) = {1, 2} × ∅
  
  = ∅
  
  Now, RHS
  
  (A × B) = {1, 2} × {1, 2, 3, 4}
  
  = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}
  
  (A × C) = {1, 2} × {5, 6}
  
  = {(1, 5), (1, 6), (2, 5), (2, 6)}
  
  Since, there is no common element between A × B and A × C
  
  (A × B) ∩ (A × C) = ∅
  
  A × (B ∩ C) = (A × B) ∩ (A × C)
  
  Answered by Sakshi | 1 year ago
  
  ### Related Questions
  
  #### Let R = {(a, b) : a, b, ϵ N and a < b}.Show that R is a binary relation on N
  
  Let R = {(a, b) : a, b, ϵ N and a < b}.Show that R is a binary relation on N, which is neither reflexive nor symmetric. Show that R is transitive.
  
  #### Let A = (1, 2, 3} and B = {4} How many relations can be defined from A to B.
  
  Let A = (1, 2, 3} and B = {4} How many relations can be defined from A to B.
  
  #### Let R = {(x, x2) : x is a prime number less than 10}.
  
  Let R = {(x, x2) : x is a prime number less than 10}.
  
  (i) Write R in roster form.
  
  (ii) Find dom (R) and range (R).
explanation: The given text contains a math question and its solution. The question is about verifying a property of Cartesian products and intersections of sets. The solution is provided, showing the step-by-step verification of the property.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that A x (B ∩ C) = (A x B) ∩ (A x C)
  
  Asked by Aaryan | 1 year ago |  55
  
  ##### Solution :-
  
  A × (B ∩ C) = (A × B) ∩ (A × C)
  
  Let us consider LHS A × (B ∩ C)
  
  (B ∩ C) = ∅
  
  A × (B ∩ C) = {1, 2} × ∅
  
  = ∅
  
  Now, RHS
  
  (A × B) = {1, 2} × {1, 2, 3, 4}
  
  = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}
  
  (A × C) = {1, 2} × {5, 6}
  
  = {(1, 5), (1, 6), (2, 5), (2, 6)}
  
  Since, there is no common element between A × B and A × C
  
  (A × B) ∩ (A × C) = ∅
  
  A × (B ∩ C) = (A × B) ∩ (A × C)
  
  Answered by Sakshi | 1 year ago
  
  ### Related Questions
  
  #### Let R = {(a, b) : a, b, ϵ N and a < b}.Show that R is a binary relation on N
  
  Let R = {(a, b) : a, b, ϵ N and a < b}.Show that R is a binary relation on N, which is neither reflexive nor symmetric. Show that R is transitive.
  
  #### Let A = (1, 2, 3} and B = {4} How many relations can be defined from A to B.
  
  Let A = (1, 2, 3} and B = {4} How many relations can be defined from A to B.
  
  #### Let R = {(x, x2) : x is a prime number less than 10}.
  
  Let R = {(x, x2) : x is a prime number less than 10}.
  
  (i) Write R in roster form.
  
  (ii) Find dom (R) and range (R).
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The given text contains a math question and its solution. The question is about verifying a property of Cartesian products and intersections of sets. The solution is provided, showing the step-by-step verification of the property.
  
  Label: question-and-answer