text: # symbols as operations
  
  • May 29th 2012, 05:08 PM
  klg
  symbols as operations
  the question is as follows:
  For all numbers x and y, let the operation ■ be defined by x ■ y = xy-y. If a and b are positive integers, which of the following can be equal to zero?
  
  I. a■b
  II. (a+b)■b
  III. a■(a+b)
  
  does one distribute choices II and III to a■b + b■b and a■a + a■b respectively? How does that help?
  There is just something about the use of symbols that has not registered yet. Thanks.
  • May 29th 2012, 06:13 PM
  Plato
  Re: symbols as operations
  Quote:
  
  Originally Posted by klg
  the question is as follows:
  For all numbers x and y, let the operation ■ be defined by x ■ y = xy-y. If a and b are positive integers, which of the following can be equal to zero?
  I. a■b
  II. (a+b)■b
  III. a■(a+b)
  
  $I~a \bullet b = ab - b$
  
  $II~(a + b) \bullet b = (a + b)b - b$
  
  $III~a \bullet (a + b) = a(a + b) - (a + b)$
  • May 29th 2012, 06:15 PM
  emakarov
  Re: symbols as operations
  Quote:
  
  Originally Posted by klg
  does one distribute choices II and III to a■b + b■b and a■a + a■b respectively?
  
  Let's see. By definition, (x + y) ■ z = (x + y)z - z = xz + yz - z, while x ■ z + y ■ z = xz - z + yz - z = xz + yz - 2z. So, ■ does not distribute over + on positive integers.
  
  A binary operator is just a function of two arguments. Expand ■ according to its definition in each expression so that you only have plus and times left.
  • May 29th 2012, 06:45 PM
  klg
  Re: symbols as operations
  Thank you very much! Now this makes sense, I just hope I can apply this the next time.
explanation: The document contains a math question and a discussion about the solution. The question involves a custom operation defined by \( x \ ■ \ y = xy - y \) and asks which of the given expressions can be equal to zero. The discussion includes a user's attempt to solve the problem and responses from other users providing guidance and clarifications.
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: # symbols as operations
  
  • May 29th 2012, 05:08 PM
  klg
  symbols as operations
  the question is as follows:
  For all numbers x and y, let the operation ■ be defined by x ■ y = xy-y. If a and b are positive integers, which of the following can be equal to zero?
  
  I. a■b
  II. (a+b)■b
  III. a■(a+b)
  
  does one distribute choices II and III to a■b + b■b and a■a + a■b respectively? How does that help?
  There is just something about the use of symbols that has not registered yet. Thanks.
  • May 29th 2012, 06:13 PM
  Plato
  Re: symbols as operations
  Quote:
  
  Originally Posted by klg
  the question is as follows:
  For all numbers x and y, let the operation ■ be defined by x ■ y = xy-y. If a and b are positive integers, which of the following can be equal to zero?
  I. a■b
  II. (a+b)■b
  III. a■(a+b)
  
  $I~a \bullet b = ab - b$
  
  $II~(a + b) \bullet b = (a + b)b - b$
  
  $III~a \bullet (a + b) = a(a + b) - (a + b)$
  • May 29th 2012, 06:15 PM
  emakarov
  Re: symbols as operations
  Quote:
  
  Originally Posted by klg
  does one distribute choices II and III to a■b + b■b and a■a + a■b respectively?
  
  Let's see. By definition, (x + y) ■ z = (x + y)z - z = xz + yz - z, while x ■ z + y ■ z = xz - z + yz - z = xz + yz - 2z. So, ■ does not distribute over + on positive integers.
  
  A binary operator is just a function of two arguments. Expand ■ according to its definition in each expression so that you only have plus and times left.
  • May 29th 2012, 06:45 PM
  klg
  Re: symbols as operations
  Thank you very much! Now this makes sense, I just hope I can apply this the next time.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a discussion about the solution. The question involves a custom operation defined by \( x \ ■ \ y = xy - y \) and asks which of the given expressions can be equal to zero. The discussion includes a user's attempt to solve the problem and responses from other users providing guidance and clarifications.
  
  Label: question-and-answer