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Questions › DSE › DSE 2009 › Q26
2 marksMathematics › Sets, relations and functions › Relations and their propertiesHardabout 4 min

DSE 2009 Question 26: Sets, relations and functions

For a set SSS, let S2=S×SS^{2}=S\times SS2=S×S. A binary relation RRR on SSS is a subset of S2S^{2}S2. RRR is transitive if (x,y)∈R(x,y)\in R(x,y)∈R and (y,z)∈R(y,z)\in R(y,z)∈R imply (x,z)∈R(x,z)\in R(x,z)∈R. RRR is negatively transitive if (x,y)∉R(x,y)\notin R(x,y)∈/R and (y,z)∉R(y,z)\notin R(y,z)∈/R imply (x,z)∉R(x,z)\notin R(x,z)∈/R. The inverse is R−1={(y,x)∣(x,y)∈R}R^{-1}=\{(y,x)\mid (x,y)\in R\}R−1={(y,x)∣(x,y)∈R}. Statement A: RRR can be transitive and negatively transitive at the same time. Statement B: If RRR is transitive then R−1R^{-1}R−1 must be transitive. Statement C: If RRR is transitive then R−1R^{-1}R−1 must be negatively transitive. How many of the above statements are true?

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For a set S, let S²=S× S. A binary relation R on S is a subset of S². R is transitive if (x,y)∈ R and (y,z)∈ R imply (x,z)∈ R. R is negatively transitive if (x,y)∉ R and (y,z)∉ R imply (x,z)∉ R. The inverse is R⁻¹={(y,x)| (x,y)∈ R}. Statement A: R can be transitive and negatively transitive at the same time. Statement B: If R is transitive then R⁻¹ must be transitive. Statement C: If R is transitive then R⁻¹ must be negatively transitive. How many of the above statements are true? (A) None (B) One (C) Two (D) Three

Also written as: For a set S, let S^2=S* S. A binary relation R on S is a subset of S^2. R is transitive if (x,y) in R and (y,z) in R imply (x,z) in R. R is negatively transitive if (x,y) not in R and (y,z) not in R imply (x,z) not in R. The inverse is R^(-1)={(y,x)| (x,y) in R}. Statement A: R can be transitive and negatively transitive at the same time. Statement B: If R is transitive then R^(-1) must be transitive. Statement C: If R is transitive then R^(-1) must be negatively transitive. How many of the above statements are true?

Sets, relations and functions has come up 65 times in 6 exams since 2004.

What this question tests

This question is on Relations and their properties, part of Sets, relations and functions in Mathematics.

Binary relations and what reflexive, symmetric, transitive, complete and order properties imply.

TestStop has 6 past-paper questions on this subtopic.

Practise Sets, relations and functionsMore on Relations and their properties

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Question from the DSE 2009 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.