homework-03 - Student Yu Cheng(Jade Math 475 Homework#3...

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Student: Yu Cheng (Jade) Math 475 Homework #3 March 1, 2010 Section 4.6 Exercise 36-a Let g be a set of G elements. How many different relations on g are there? Answer: On set g with G elements, we have the following facts. Number of two different element pairs ± ² ³ ´ Number of relations on two different elements µ¶, ·¸ ∈ ¹, µ·, ¶¸ ∈ ¹ 2 º ± ² ³ ´ Number of relations including the reflexive ones µ¶,¶¸ ∈ ¹ 2 º ± ² ³ ´ » G Number of ways to select these relations to form a relation on g 2 ³º± ¼ ½ ´¾² 2 ³º± ¼ ½ ´¾² ¿ 2 ³º²! µ²À³¸!º³ ¾² ¿ 2 ²µ²ÀÁ¸¾² ¿ 2 ² ½ . b. How many of these relations are reflexive? Answer: We still have 2 º ± ² ³ ´ » G number of relations on element pairs to choose from, but we have to include the reflexive one, µ¶, ¶¸ ∈ ¹ . There are G relations of this kind. Therefore there are 2 ±³º± ¼ ½ ´¾²´À² ¿ 2 ³º± ¼ ½ ´ ¿ 2 ³º²! µ²À³¸!º³ ¿ 2 ²µ²ÀÁ¸ . c. How many of these relations are symmetric? Answer: To select only the symmetric relations on set g with G elements, we have the following facts. Number of symmetric relation pairs between two elements ± ² ³ ´ Number of relations including the reflexive ones ± ² ³ ´ » G Number of ways to select these relations to form a relation on g 2 ± ¼ ½ ´¾² 2 ± ¼ ½ ´¾² ¿ 2 ²! µ²À³¸!º³ ¾² ¿ 2 ²µ²ÀÁ¸ ³ ¾² ¿ 2 ²µ²¾Á¸ ³ .
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d. How many of these relations are anti-symmetric? Answer: To select only the anti-symmetric relations on set g with G elements, we have the following facts. Number of anti-symmetric relations between two elements ± ² ³ ´ Number of relations including the reflexive ones µ¶,¶· ∈ ¸ ± ² ³ ´ ¹ G Number of ways to select these relations to form a relation on g 2 ± º » ´¼² 2 ± º » ´¼² ½ 2 ²! µ²¾³·!¿³ ¼² ½ 2 ²µ²¾À· ³ ¼² ½ 2 ²µ²¼À· ³ . e. How many of these relations are reflexive and symmetric? Answer: To select only the reflexive and symmetric relations on set g with G elements, we have the following facts. Number of symmetric and reflexive relations between two elements ± ² ³ ´ Number of ways to select these relations to form a relation on g 2 ± º » ´ 2 ± º » ´ ½ 2 ²! µ²¾³·!¿³ ½ 2 ²µ²¾À· ³ . f. How many of these relations are reflexive and anti-symmetric? Answer: Subtracting the number of relations on g that are reflexive from the number of relations on g that are anti-symmetric would result the number of relations on g that are reflexive and anti- symmetric. 2 ± º » ´ ½ 2 ²! µ²¾³·!¿³ ½ 2 ²µ²¾À· ³ . Exercise 37 Let ¸′ and ¸" be two partial orders on a set g . Define a new relation ¸ on g by Á¸Â if and only if both Á¸′ and Á¸" hold. Prove that ¸ is also a partial order on g . ( ¸ is called the intersection of ¸′ and ¸" .) Answer: According to the definition of partial order set, we need to prove three properties on the given relation ¸ : reflexive, transitive, and anti-symmetric hold.
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