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Chapter 0: Problem 10

\(A=\\{a, b, c\\}\) and \(B=\\{0,1,2,3\\}\) (a) \(\\{(a, 1),(c, 2),(c, 3),(b, 3)\\}\) (b) \(\\{(a, 1),(b, 2),(c, 3)\\}\) (c) \(\\{(1, a),(0, a),(2, c),(3, b)\\}\) (d) \(\\{(c, 0),(b, 0),(a, 3)\\}\)

Short Answer

Expert verified
(a) is not any type of relation and not a function. (b) is a function. (c) is not any type of relation and not a function. (d) is a function.

Step by step solution

01

Evaluate (a)

For (a) \{(a, 1),(c, 2),(c, 3),(b, 3)\}, we can see that it is not reflexive (since not all elements relate to themselves), not symmetric (the pairs are not mirrored), not antisymmetric (none of the mirrored pairs exist), and not transitive (there is no pair that leads to another). It is not a function since 'c' is related to both '2' and '3'.
02

Evaluate (b)

For (b) \{(a, 1),(b, 2),(c, 3)\}, we can observe that it is not reflexive (since not all elements relate to themselves), not symmetric (none of the pairs are mirrored), not antisymmetric (none of the mirrored pairs exist), and not transitive (there is no pair that leads to another). It is a function since every element in A is related to exactly one element in B.
03

Evaluate (c)

For (c) \{(1, a),(0, a),(2, c),(3, b)\}, we notice that all sets are not symmetric, not antisymmetric, and not transitive. Additionally, it is not a function because 'a' does not map to exactly one element in B.
04

Evaluate (d)

For (d) \{(c, 0),(b, 0),(a, 3)\}, similar to the previous sets, it fails all relation conditions. It can be considered as a function, as every element in A is related to one element in B.

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Most popular questions from this chapter

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