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Relations

Building relations as sets of ordered pairs between finite sets, counting how many are possible, and testing notions like being an endorelation.

A Cool Brisk Walk Through Discrete Mathematics, Relations

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38 exercises

Exercise 1

Let $A$ be the set { Chuck, Julie, Sam } and $S$ be the set { basketball, volleyball }. Is { (Julie, basketball), (Sam, basketball), (Julie, volleyball) } a relation between $A$ and $S$ ?

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Exercise 2

Is the above relation an endorelation?

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Exercise 3

Is { (Chuck, basketball), (basketball, volleyball) } a relation between $A$ and $S$ ?

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Exercise 4

Is $\varnothing$ a relation between $A$ and $S$ ?

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Exercise 5

How large could a relation between $A$ and $S$ be?

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Exercise 6

Let $T$ be the set { Spock, Kirk, McCoy, Scotty, Uhura }. Let $O$ be an endorelation on $T$ , defined as follows: { (Kirk, Scotty), (Spock, Scotty), (Kirk, Spock), (Scotty, Spock) }. Is $T$ reflexive?

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Exercise 7

Is $T$ symmetric?

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Exercise 8

Is $T$ antisymmetric?

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Exercise 9

Is $T$ transitive?

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Exercise 10

Let $H$ be an endorelation on $T$ , defined as follows: { (Kirk, Kirk), (Spock, Spock), (Uhura, Scotty), (Scotty, Uhura), (Spock, McCoy), (McCoy, Spock), (Scotty, Scotty), (Uhura, Uhura) }. Is $H$ reflexive?

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Exercise 11

Is $H$ symmetric?

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Exercise 12

Is $H$ antisymmetric?

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Exercise 13

Is $H$ transitive?

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Exercise 14

Let outranks be an endorelation on the set of all crew members of the Enterprise, where $(x,y) \in$ outranks if character $x$ has a higher Star Fleet rank than $y$ . Is outranks reflexive?

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Exercise 15

Is outranks symmetric?

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Exercise 16

Is outranks antisymmetric?

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Exercise 17

Is outranks transitive?

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Exercise 18

Is outranks a partial order?

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Exercise 19

Let sameShirtColor be an endorelation on the set of all crew members of the Enterprise, where $(x,y) \in$ sameShirtColor if character $x$ ordinarily wears the same shirt color as character $y$ . Is sameShirtColor reflexive?

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Exercise 20

Is sameShirtColor symmetric?

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Exercise 21

Is sameShirtColor antisymmetric?

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Exercise 22

Is sameShirtColor transitive?

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Exercise 23

Above, we defined $A$ as the set { Chuck, Julie, Sam } and $S$ as the set { basketball, volleyball }. Then we defined the relation { (Julie, basketball), (Sam, basketball), (Julie, volleyball) }. Is this relation a function?

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Exercise 24

Suppose we added the ordered pair (Chuck, basketball) to it. Now is it a function?

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Exercise 25

Okay. Suppose we then remove (Julie, volleyball). We now have { (Julie, basketball), (Sam, basketball), (Chuck, basketball) }. Is this a function?

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Exercise 26

Let's call this function “faveSport,” which suggests that its meaning is to indicate which sport is each athlete's favorite. What's the domain of faveSport?

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