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Understanding Relations and Equivalence in Discrete Mathematics

This guide provides a comprehensive overview of key concepts in discrete mathematics, focusing on projections, combinations of relations, and equivalence relations. Through sample questions, it explains the meaning of projections, how to combine relations, and how to prove a relation is an equivalence relation, along with determining the number of equivalence classes. Whether you are preparing for exams or looking to solidify your understanding, this resource offers clarity on essential terms and definitions used in the study of discrete mathematics.

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Understanding Relations and Equivalence in Discrete Mathematics

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  1. Discrete MathematicsReview Question September 24

  2. Sample Question 1 • Terms and Definitions *There is no need for you to remember everything but just understand what it means Example: What’s the meaning of Projection? A. Function that combines n-ary relations that agree on certain fields B. Function that conditionally picks n-tuples from a relation C. Function that produces relations of smaller degree from an n-aryrelation

  3. Sample Question 2 • Combining Relations *Give you relations R1 and R2. Need you to define like … Example: R1 = {(a,b) on R2 | a > b} R2 = {(a,b) on R2| a < b}  {(a,b) on R2| a != b}  R2 For (a, c) to be in R1 R2 , we must find an element b such that (a, b) in R2 and (b, c) in R1 . This means that a < b and b > c. Clearly this can always be done simply by choosing b to be large enough. Therefore we have R1 R2 = R2 , the relation that always holds.

  4. Sample Question 3 • Equivalence proof *Give you a relation, proof why it is a equivalence relation and tell how many equivalence class it has. Example: Suppose we have R ={(a,b) | a b (mod 3)} on set Z+. Proof R is an equivalence relation. * Just proof the reflexive, symmetric and transitive of R. (Text book Section 9.5 Example 3) How many equivalence class it has? * Clearly, 3 ( divided by 3 remain 1,2,0). (Text book Section 9.5 Example 14)

  5. Sample Question 4 • Matrix *Give you a relation represented by matrix, proof why it is a equivalence relation Example: We have a matrix representation of a relation like: 1 2 3 1 [1 0 1] Is this relation a equivalence relation? 2 [0 1 0] List all equivalence class. 3 [1 0 1]

  6. Ricky Gao email gxr116020@utdallas.edu • TA Office hour Mon & Wed 9:00am – 10:00am at ECSS 3.216

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