True/False (Write the word True or False in the blanks.) 2pts each. Section 4.1

Question

True/False (Write the word True or False in the blanks.) 2pts each.

Section 4.1 Relations

________1. In a one-to-many binary relation, at least one second component must appear in two different ordered pairs.

________2. The following is a many-to-one binary relation: {(2,1), (2,2)}

________3. A relation from S to T must contain every element of S as a first component.

________4. S = {4, 5, 6}. The following is a reflexive binary relation on S:

{(4,5), (5,6), (4,4), (5,4), (6,6), (6,4), (5,5)}

________5. S = {4, 5, 6}. The following is a symmetric binary relation on S:

{(4,5), (5,6), (4,4), (5,4), (5,5), (6,6)}

________6. S = {4, 5, 6}. The following is a transitive binary relation on S:

{(4,5), (5,6), (4,6), (3,4)}

________7. S = {0, 1}. The following is an antisymmetric binary relation on S:

{(0,0), (1,1)}

________8. A binary relation on S that is reflexive, antisymmetric, and transitive is called a partial ordering.

________9. A binary relation on a set S that is reflexive, symmetric, and transitive is called an equivalence relation on S.

________10. Predecessors and successors are important features of partial orderings.

________11. Equivalence relations determine partitions and partitions determine equivalence relations.

Section 4.4 Functions

________12. An onto function means that every element in the domain must have an image.

________ 13. An onto function means that every element in the codomain must have an image.

________ 14. An onto function means that every element in the codomain must have a preimage.

________ 15. An onto function means that every element in the codomain must have a unique preimage.

________ 16. A one-to-one function means that every element in the codomain must have a unique preimage.

________ 17. A one-to-one function means that no two elements in the domain map to the same element in the codomain.

________ 18. An onto function means that (the range) ? (the codomain) =

Explanation / Answer

____T____1. In a one-to-many binary relation, at least one second component must appear in two different ordered pairs.

____T____2. The following is a many-to-one binary relation: {(2,1), (2,2)}

____T____3. A relation from S to T must contain every element of S as a first component.

____F____4. S = {4, 5, 6}. The following is a reflexive binary relation on S:

{(4,5), (5,6), (4,4), (5,4), (6,6), (6,4), (5,5)}

____F____5. S = {4, 5, 6}. The following is a symmetric binary relation on S:

{(4,5), (5,6), (4,4), (5,4), (5,5), (6,6)}

____T____6. S = {4, 5, 6}. The following is a transitive binary relation on S:

{(4,5), (5,6), (4,6), (3,4)}

___T_____7. S = {0, 1}. The following is an antisymmetric binary relation on S:

{(0,0), (1,1)}

____T____8. A binary relation on S that is reflexive, antisymmetric, and transitive is called a partial ordering.

____T____9. A binary relation on a set S that is reflexive, symmetric, and transitive is called an equivalence relation on S.

____T____10. Predecessors and successors are important features of partial orderings.

____T____11. Equivalence relations determine partitions and partitions determine equivalence relations.

Section 4.4 Functions

____F____12. An onto function means that every element in the domain must have an image.

____T____ 13. An onto function means that every element in the codomain must have an image.

____T____ 14. An onto function means that every element in the codomain must have a preimage.

____F____ 15. An onto function means that every element in the codomain must have a unique preimage.

____T____ 16. A one-to-one function means that every element in the codomain must have a unique preimage.

____F____ 17. A one-to-one function means that no two elements in the domain map to the same element in the codomain.

____T___ 18. An onto function means that (the range) ? (the codomain) =

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