A={a,b,c,d}

X={1,2,3,4}

Select the definition for:

f:A -- X that is a well-defined total function.

Question

A={a,b,c,d}

X={1,2,3,4}

Select the definition for:

f:A --> X that is a well-defined total function.

BlackTom AI · Accepted Answer

A function f: A -> X must assign to every element of the domain A exactly one element of the codomain X. That means (1) every element a, b, c, d must appear as the first component of exactly one ordered pair (or at least once with the same image), and (2) every second component must be an element of X = {1,2,3,4}. Also the ordered-pair format must be (domain-element, codomain-element).

1) Relevant concepts
- Domain coverage (total function): each element of A must have a mapping.
- Well-defined / single-valued: each domain element maps to a unique codomain element.
- Ordered-pair orientation: first entry must be from A, second from X.

2) Step-by-step checking of options
Option 1: f = {(a,2),(b,3),(d,1)}  
- Domain elements present: a, b, d. Missing c → not total. Reject.

Option 2: f = {(a,2),(b,3),(b,3),(d,1)}  
- Domain elements present: a, b, d (b is repeated). c is missing → not total. Repeated identical pair is redundant but does not supply c. Reject.

Option 3: f = {(a,2),(b,3),(c,3),(d,1)}  
- Domain elements present: a, b, c, d — all covered.  
- Images: 2, 3, 3, 1 are all in X = {1,2,3,4}.  
- Each domain element has exactly one image (c and b both map to 3 is allowed). Satisfies total and well-defined. Accept.

Option 4: f = {(a,2),(b,3),(c,3),(1,d)}  
- The pair (1,d) has first component 1, which is not in A (1 ∈ X). Also d (second component) is in A, not in X. This violates ordered-pair orientation and codomain membership. Reject.

3) Why the correct answer is correct
Option 3 assigns every element of A a single value in X and uses properly ordered pairs. It meets both totality and well-definedness.

4) Why other options are incorrect
Options 1 and 2 omit c from the domain mapping. Option 4 has an incorrectly oriented pair and uses elements outside the required positions.

Conclusion: The correct choice is f = {(a,2),(b,3),(c,3),(d,1)}, matching the provided answer.

A={a,b,c,d} X={1,2,3,4} Select the definition for: f:A --> X that is a well-defined total function.单项选择题

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