32.7 Equivalent Representation Verification

Equivalent Representation Verification is the systematic process of confirming that two given representations of a function — whether a table, a rule, an ordered pair list, a mapping diagram, or a graph — actually describe the exact same set of input-output associations, by checking that both share the same inputs, assign each input the same output, and contain no missing or extra points relative to one another. This process extends the general idea introduced under same relation in different forms into a precise, checkable procedure specifically suited to comparing any two representations encountered while working across multiple forms of a function.

Because two representations can look quite different on the surface while still describing the same function, or can look deceptively similar while actually differing in a small but important detail, this verification process relies on breaking each representation down into its underlying input-output pairs before any comparison is made.

Shared Function Input Set

Extracting the Input Set From Each Representation

The first step in verification is extracting the full set of input values from each representation being compared, using the appropriate extraction method described under finite domain identification for whichever specific form, whether table, diagram, or graph, is involved.

Comparing the Two Extracted Input Sets

Once both input sets have been extracted, they are compared directly to confirm that they contain exactly the same values, with neither set including an input the other one lacks.

Why a Shared Input Set Is a Necessary Starting Point

If the two representations do not even agree on which inputs are involved, they cannot possibly describe the same function, making agreement on the input set a necessary first checkpoint before any comparison of outputs is attempted.

Shared Output Assignment

Comparing Outputs for Each Shared Input

For every input confirmed to be present in both representations, the output assigned to that input in the first representation is compared against the output assigned to that same input in the second representation.

Confirming Consistent Assignment Across Both Forms

Agreement is confirmed only when every shared input produces the identical output in both representations, following the same input-output association standard described under common input-output associations.

An Example of Comparing Assignments

If a table assigns the input $$3$$ to the output $$9$$, and a candidate rule assigns that same input to a different output when evaluated, this specific input reveals a disagreement between the two representations rather than a confirmation of equivalence.

Missing Function Point Check

Identifying a Point Present in One Form but Absent From Another

A missing point occurs when an input-output pair appears in one representation but has no corresponding entry at all in the other, meaning that specific pairing is simply absent rather than merely disagreeing with a different recorded output.

Locating Missing Points Systematically

Locating missing points involves comparing the full inventory of pairs from each representation side by side, checking specifically for any pair present in one inventory that has no counterpart in the other.

Consequences of a Missing Point for Equivalence

A single missing point is enough to establish that the two representations are not fully equivalent, since one of them describes an association that the other simply does not include at all.

Extra Function Point Check

Identifying a Point Present in One Form That Should Not Be There

An extra point occurs when a representation includes an input-output pair that does not correspond to anything in the other representation being compared against, effectively adding information beyond what the two forms are supposed to share.

Locating Extra Points Systematically

Locating extra points follows the same side-by-side comparison used to locate missing points, but from the opposite direction, checking whether either inventory contains a pair with no counterpart in the other.

Consequences of an Extra Point for Equivalence

As with a missing point, a single extra point is enough to establish that the two representations are not equivalent, since the representation containing the extra point describes more associations than the one it is being compared against.

Domain Preservation across Forms

Confirming the Domain Matches in Both Representations

Beyond checking individual points, the overall domain extracted from each representation, following finite domain identification, is compared to confirm that both representations describe a function over exactly the same set of allowed inputs.

Why Domain Preservation Is a Summary Check

Because the domain is built directly from the individual input values already checked during the shared input set step, confirming domain preservation functions as a useful summary check that consolidates those individual checks into a single, clear comparison.

Detecting a Domain Mismatch

A domain mismatch, where one representation's domain set differs from the other's, directly signals that the two representations are not equivalent, even without needing to examine every individual pair again in detail.

Range Preservation across Forms

Confirming the Range Matches in Both Representations

Similarly, the overall range extracted from each representation, following finite range identification, is compared to confirm that both representations produce exactly the same set of output values.

Why Range Preservation Complements Domain Preservation

While domain preservation confirms that both representations involve the same inputs, range preservation confirms that both representations produce the same overall collection of outputs, together covering both halves of what a fully equivalent representation must share.

Detecting a Range Mismatch

A range mismatch, where one representation's range set differs from the other's, similarly signals non-equivalence, and combined with a domain mismatch or an individual missing or extra point, helps pinpoint exactly where and how the two representations diverge.

Function Representation Equivalence Decision

Confirming Full Equivalence

Two representations are confirmed as equivalent once every check has passed: the input sets match, every shared input produces the same output in both, no points are missing from either side, and no extra points appear in either side.

Declaring Non-Equivalence

If any single check fails, whether a mismatched output, a missing point, an extra point, or a domain or range mismatch, the two representations are declared non-equivalent, and the specific failed check provides the evidence supporting that conclusion.

Reporting the Equivalence Decision With Evidence

A completed equivalence verification is reported together with the specific evidence supporting the decision, whether a full statement that every check passed for a confirmed equivalence or the precise point of disagreement for a declared non-equivalence, keeping the conclusion checkable rather than asserted without support.