What if one input has two answers?
Distinguish exactly one output per input from missing or split assignments.
Begin with the activity ↓What if one input has two answers?
Assign a locker number to each card numbered 0, 1, 2 and 3. If card 1 points to both locker 2 and locker 3, where should you go? One chosen input needs one determined output for the instruction to be unambiguous.
Assign a locker number to each card numbered 0, 1, 2 and 3. If card 1 points to both locker 2 and locker 3, where should you go? One chosen input needs one determined output for the instruction to be unambiguous.
Call the card number the input and the locker number the output. A function assigns exactly one output to every allowed input. No allowed input may be missing an output or split between two different outputs.
Different cards, such as 1 and 2, may both point to locker 2. Looking at each card separately, its output is still unique. A function does not require different inputs to have different outputs.
The assignments 0→1, 1→2, 2→2 and 3→3 form a function. Adding 1→3 while keeping 1→2 gives input 1 two outputs. Removing the assignment for 3 while still allowing 3 leaves an input without an output.
The lockers illustrate a mapping rather than a real allocation policy. If a machine changes with time or state, fix those conditions or include them as inputs. A real machine without stated conditions is not automatically a mathematical function.
Change the assignments below and trace one input at a time. Arrows merging and an input splitting are different. Next we specify exactly which card numbers are allowed inputs.
Listen one paragraph at a time
Hold on to this distinction
Several arrows may meet at an output. Start from each input and count its distinct outputs. Zero or more than one fails the function condition for an allowed input.
Change it and check
Choose merged, split or missing assignments. Predict whether it is a function, then select each input and compare its arrows with its table row.
Domain: 0, 1, 2, 3 · Codomain: 1, 2, 3
| Input | Output |
|---|---|
| 0 | 1 |
| → 1 | 2 |
| 2 | 2 |
| 3 | 3 |
1 → 2
Function: each allowed input has exactly one output.
Reached outputs: 1, 2, 3 · Range
Allowed inputs are 0, 1, 2 and 3; outputs belong to 1, 2 and 3. The diagram shows this entire finite relation.
Try with paper and objects
Write cards 0–3 on the left and lockers 1–3 on the right. Draw both 1 and 2 to locker 2, then explain what changes if input 1 gets an additional output.
Check and explain another example
Check your idea 1
Check your idea 2
Apply it in a new setting
Write an answer yourself
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3 · distinct outputs
The distinct outputs are 1, 2 and 3. Count repeated output 2 once.
Does your explanation include these ideas?
Explain aloud or in a drawing, then check your ideas. Different wording can express the same meaning.
Continue, or pause here
Distinguish allowed inputs, the output set and the outputs actually reached.