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formalize a two-place relation over a declared domain
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will formalize a two-place relation over a declared domain, recording the intermediate model values and the precise reason each conclusion follows.
Recall the truth conditions for not, and, or and if-then. Those rules still govern compound formulas here, but atomic truth now comes from objects and predicate extensions or from a world's valuation. Identify which new structure this lesson introduces before using a familiar propositional rule.
| Term | What it means |
|---|---|
| Interpretation | A declared domain and meanings for the nonlogical symbols; a modal interpretation also specifies worlds, accessibility and valuations. |
| Assignment | A choice of domain object for a free variable during an evaluation; it is not itself another domain object. |
| Witness | An eligible object or accessible world satisfying the property required by an existential or possibility claim. |
| Counterexample | An eligible case where the required condition fails; a countermodel to an inference additionally makes every premise true. |
| Validity | Truth in every interpretation of the specified kind, a stronger claim than truth in one supplied model. |
A two-place predicate represents a relation between ordered pairs of objects. R(a,b) places the denotation of a in the first argument and the denotation of b in the second. If R means sends a message to, reversing the arguments changes who sends and who receives. Neither direction is implied by the other unless the interpretation or an explicit premise supplies it.
The extension is a set of ordered pairs. For a finite domain, display the first argument down the rows and the second across columns. A true cell records a listed pair; a false cell records an unlisted pair in the complete interpretation. The diagonal contains pairs whose two arguments are the same object. Self-pairs are legitimate syntactically even when an intended relation happens never to contain one.
A relation is symmetric if every true pair has its reverse. It is reflexive if every self-pair is true. It is transitive if R(a,b) and R(b,c) always bring R(a,c). These properties are independent requirements, not automatic features of a two-place predicate. One pair with its reverse does not establish symmetry for the whole domain.
Be equally careful in translation. If the key says R(x,y) means x borrows from y, 'Ada lends to Bo' becomes R(Bo,Ada). Follow the stated roles rather than the order of names in English. The worksheets assess interpretation of complete relations; a sparse real record of messages or loans would require care before treating every absent entry as false.
Another way: An explicit audit sheet
Keep four parts on the page: the declared objects or worlds, the meaning of each symbol, the intermediate values, and the conclusion. A changed interpretation belongs on a new sheet so the premises and conclusion are never checked in different models.
Domain: a, b, all distinct. R holds exactly for these ordered pairs: (a,a), (b,b). Every unlisted pair is false, and self-pairs are permitted. Give R(a,b), R(b,a), and the number of true self-pairs R(x,x).
Fix the first argument at a. Its row has targets a, so there are 1 true entries in this row. The statement exists y R(a,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Fix the first argument at b. Its row has targets b, so there are 1 true entries in this row. The statement exists y R(b,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
R(a,b): F. Read the first name as the source and the second as the target. Check the ordered pair in that direction against the extension. A binary predicate records a property of an ordered pair, so reversing its arguments can change its truth. The interpretation supplies all and only the true pairs; there is no hidden assumption that a relation behaves like friendship or equality.
R(b,a): F. Inspect the reverse ordered pair independently. Its truth is not inherited from the first answer unless symmetry was supplied or established for the whole relation. A pair can occur without its reverse. A model may also contain both directions, in which case both atomic formulas are true for that particular pair; that local fact alone still does not establish universal symmetry.
True self-pairs: 2. Look along the diagonal where the two arguments name the same object. Count each listed self-pair once. Some relations permit self-relations and some do not, but neither behavior follows merely from being a relation. Reflexivity requires every domain object to appear on this diagonal. A nonzero diagonal count is therefore weaker than a proof that the entire relation is reflexive.
Transpose the table by exchanging the first and second positions in every pair. The transpose describes the converse relation, not automatically the original relation. Its row for an object lists sources that previously pointed to that object. Comparing a relation with its transpose is one way to test symmetry. Keep this operation separate from negation, which changes true entries to false and false entries to true over the declared domain.
Domain: a, b, c, all distinct. R holds exactly for these ordered pairs: (a,c), (b,c), (c,c). Every unlisted pair is false, and self-pairs are permitted. Give R(a,b), R(b,a), and the number of true self-pairs R(x,x).
Fix the first argument at a. Its row has targets c, so there are 1 true entries in this row. The statement exists y R(a,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Fix the first argument at b. Its row has targets c, so there are 1 true entries in this row. The statement exists y R(b,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Fix the first argument at c. Its row has targets c, so there are 1 true entries in this row. The statement exists y R(c,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
R(a,b): F. Read the first name as the source and the second as the target.
R(b,a): F. Inspect the reverse ordered pair independently.
True self-pairs: 1. Look along the diagonal where the two arguments name the same object.
Transpose the table by exchanging the first and second positions in every pair. The transpose describes the converse relation, not automatically the original relation. Its row for an object lists sources that previously pointed to that object. Comparing a relation with its transpose is one way to test symmetry. Keep this operation separate from negation, which changes true entries to false and false entries to true over the declared domain.
Domain: a, b, c, d, all distinct. R holds exactly for these ordered pairs: (a,b), (b,c), (c,d), (d,a). Every unlisted pair is false, and self-pairs are permitted. Give R(a,b), R(b,a), and the number of true self-pairs R(x,x).
Fix the first argument at a. Its row has targets b, so there are 1 true entries in this row. The statement exists y R(a,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Fix the first argument at b. Its row has targets c, so there are 1 true entries in this row. The statement exists y R(b,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Fix the first argument at c. Its row has targets d, so there are 1 true entries in this row. The statement exists y R(c,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Fix the first argument at d. Its row has targets a, so there are 1 true entries in this row. The statement exists y R(d,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
R(a,b): T. Read the first name as the source and the second as the target.
R(b,a): F. Inspect the reverse ordered pair independently.
True self-pairs: 0. Look along the diagonal where the two arguments name the same object.
Transpose the table by exchanging the first and second positions in every pair. The transpose describes the converse relation, not automatically the original relation. Its row for an object lists sources that previously pointed to that object. Comparing a relation with its transpose is one way to test symmetry. Keep this operation separate from negation, which changes true entries to false and false entries to true over the declared domain.
A relation's extension lists exactly the pairs it contains. If a relates to b and b relates to c, the pair from a to c is not automatically included. That additional requirement is transitivity. A direct-message relation is usually not transitive: sending a message to someone who sends a message onwards does not mean the original sender sent directly to the final recipient.
A reachability relation can be built from direct links by admitting paths. This operation changes the relation. A path of two links may establish reachability even when the corresponding direct pair is absent. If a problem asks for the direct relation, adding all reachable pairs answers a different question. Mark the predicate key before constructing a graph.
Reflexivity introduces another distinction. A path of length zero can be allowed in a definition of reachability, making every object reachable from itself. A definition requiring a nonempty path need not have that property unless a cycle returns to the starting object. These conventions must be stated rather than guessed from the word reachable.
The reverse relation is yet another operation. R-converse(a,b) holds exactly when R(b,a) holds. A relation is symmetric when it equals its converse. It is not enough to find one two-way connection; every true pair must have the reverse. A small directed cycle is a helpful test case: it has a route back to each starting point but can lack every immediate reverse pair. Distinguishing reverse links, paths and self-links keeps three different structural properties from collapsing into a vague impression that the graph is connected.
A workshop records these transfers: Ada passed a tool to Bo, Bo passed one to Cy, and Cy passed one to Ada. Let T(x,y) mean x passed a tool to y during the recorded session. The extension contains the three ordered pairs in exactly those directions. T(Ada,Bo) is true, but T(Bo,Ada) is false in this complete record.
There is a cycle, but the relation is not symmetric. A cycle lets you travel back to a starting point through several links; symmetry requires an immediate reverse for each link. Likewise, the first two transfers do not establish that Ada passed a tool directly to Cy. The relation concerns direct recorded transfers, not whether a tool could travel through an intermediate person.
Changing the predicate to can reach through a chain of transfers would produce a different relation. That new relation might include Ada-to-Cy even when the direct-transfer relation does not. The difference lies in the meaning of the predicate, so it cannot be introduced silently halfway through an argument.
To audit the record, write the participant domain and a table with senders as rows. Put one true cell at each listed transfer and keep the diagonal false because no self-transfer is recorded. Reading a row answers whom that person supplied. Reading a column answers who supplied that person. The two questions use the same table in different directions. This simple discipline prevents a formal relation from inheriting convenient everyday assumptions that its stated meaning does not warrant.
A correct evaluation answers the stated question for its stated interpretation. Do not turn a true instance into a universal rule or a successful example into a proof of validity. When the task is a proof audit, keep local assumptions and fresh parameters within their declared scope.
Record the interpretation and the question.
Domain: a, b, c, all distinct. R holds exactly for these ordered pairs: (a,a), (a,b), (a,c), (b,a), (b,b), (b,c). Every unlisted pair is false, and self-pairs are permitted. Give R(a,b), R(b,a), and the number of true self-pairs R(x,x).
Use the declared objects and meanings throughout this calculation: R(a,b) is the first requested result.
Determine the requested value: R(a,b).
T
Read the first name as the source and the second as the target.
Determine the requested value: R(b,a).
T
Inspect the reverse ordered pair independently.
Determine the requested value: True self-pairs.
2
Look along the diagonal where the two arguments name the same object.
Collect the results in the requested order.
T / T / 2
Each result belongs to its own entry: R(a,b); R(b,a); True self-pairs.
Record the interpretation and the question.
Domain: a, b, c, d, all distinct. R holds exactly for these ordered pairs: (a,a), (b,b), (c,c), (d,d). Every unlisted pair is false, and self-pairs are permitted. Give R(a,b), R(b,a), and the number of true self-pairs R(x,x).
Use the declared objects and meanings throughout this calculation: R(a,b) is the first requested result.
Determine the requested value: R(a,b).
F
Read the first name as the source and the second as the target.
Determine the requested value: R(b,a).
F
Inspect the reverse ordered pair independently.
Determine the requested value: True self-pairs.
4
Look along the diagonal where the two arguments name the same object.
Collect the results in the requested order.
F / F / 4
Each result belongs to its own entry: R(a,b); R(b,a); True self-pairs.
Record the interpretation and the question.
Domain: a, b, c, d, e, all distinct. R holds exactly for these ordered pairs: (a,e), (b,e), (c,e), (d,e), (e,e). Every unlisted pair is false, and self-pairs are permitted. Give R(a,b), R(b,a), and the number of true self-pairs R(x,x).
Use the declared objects and meanings throughout this calculation: R(a,b) is the first requested result.
Determine the requested value: R(a,b).
F
Read the first name as the source and the second as the target.
Determine the requested value: R(b,a).
F
Inspect the reverse ordered pair independently.
Determine the requested value: True self-pairs.
1
Look along the diagonal where the two arguments name the same object.
Collect the results in the requested order.
F / F / 1
Each result belongs to its own entry: R(a,b); R(b,a); True self-pairs.
Test which alteration would change the conclusion.
Transpose the table by exchanging the first and second positions in every pair. The transpose describes the converse relation, not automatically the original relation. Its row for an object lists sources that previously pointed to that object. Comparing a relation with its transpose is one way to test symmetry. Keep this operation separate from negation, which changes true entries to false and false entries to true over the declared domain.
The altered interpretation checks the dependence of these answers on the stated model, rather than replacing it during the calculation.
Determine the requested value: R(a,b).
T
Read the first name as the source and the second as the target.
Determine the requested value: R(b,a).
Determine the requested value: True self-pairs.
Domain: a, b, c, d, e, all distinct. R holds exactly for these ordered pairs: (a,a), (a,b), (a,c), (a,d), (a,e), (b,a), (b,b), (b,c), (b,d), (b,e), (c,a), (c,b), (c,c), (c,d), (c,e), (d,a), (d,b), (d,c), (d,d), (d,e). Every unlisted pair is false, and self-pairs are permitted. Give R(a,b), R(b,a), and the number of true self-pairs R(x,x).
| Computed result | |
|---|---|
| R(a,b) | |
| R(b,a) | |
| True self-pairs |
Domain: a, b, c, d, e, f, g, all distinct. R holds exactly for these ordered pairs: (a,a), (a,b), (a,c), (a,d), (a,e), (a,f), (a,g), (b,a), (b,b), (b,c), (b,d), (b,e), (b,f), (b,g), (c,a), (c,b), (c,c), (c,d), (c,e), (c,f), (c,g), (d,a), (d,b), (d,c), (d,d), (d,e), (d,f), (d,g), (e,a), (e,b), (e,c), (e,d), (e,e), (e,f), (e,g), (f,a), (f,b), (f,c), (f,d), (f,e), (f,f), (f,g). Every unlisted pair is false, and self-pairs are permitted. Give R(a,b), R(b,a), and the number of true self-pairs R(x,x).
Inspect the pair in the displayed direction.
b0
The requested entry concerns r(a,b); retain its stated scope.
Inspect the reversed pair independently.
b1
The requested entry concerns r(b,a); retain its stated scope.
Count the listed diagonal pairs.
b2
The requested entry concerns true self-pairs; retain its stated scope.
Domain a, b, c. The complete extension of R is {(a,b), (b,c), (c,a)}. Construct its directed relation graph, with the first argument as source. Include exactly the listed pairs.
This task has no paper form; do it on a device.
Domain: a, b, c, d, e, all distinct. R holds exactly for these ordered pairs: (a,e), (b,e), (c,e), (d,e), (e,e). Every unlisted pair is false, and self-pairs are permitted. Give R(a,b), R(b,a), and the number of true self-pairs R(x,x).
R(a,b): b0
R(b,a): b1
True self-pairs: b2
Domain: a, b, c, d, e, f, all distinct. R holds exactly for these ordered pairs: (a,b), (b,c), (c,d), (d,e), (e,f), (f,a). Every unlisted pair is false, and self-pairs are permitted. Give R(a,b), R(b,a), and the number of true self-pairs R(x,x).
R(a,b): b0
R(b,a): b1
True self-pairs: b2
In a workshop transfer log, R(x,y) means x passed a tool directly to y during the session. The data below form the complete invented audit. Domain: a, b, c, d, e, f, all distinct. R holds exactly for these ordered pairs: (a,a), (b,b), (c,c), (d,d), (e,e), (f,f). Every unlisted pair is false, and self-pairs are permitted. Give R(a,b), R(b,a), and the number of true self-pairs R(x,x).
R(a,b): b0
R(b,a): b1
True self-pairs: b2
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Domain: a, b, c, d, e, f, g, all distinct. R holds exactly for these ordered pairs: (a,g), (b,g), (c,g), (d,g), (e,g), (f,g), (g,g). Every unlisted pair is false, and self-pairs are permitted. Give R(a,b), R(b,a), and the number of true self-pairs R(x,x).
R(a,b): b0
R(b,a): b1
True self-pairs: b2
You can formalize a two-place relation over a declared domain. Reconstruct the three audit entries from a fresh model without consulting the examples; explain what change to the interpretation would change one answer.
14. Complete the next model audit, step 2
F
Inspect the reverse ordered pair independently.
14. Complete the next model audit, step 3
0
Look along the diagonal where the two arguments name the same object.