# Question 15815: A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess [Return to catalogue page 80](https://ocme.evemisslab.com/en/questions/page/80/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261031-r01-at-most-countable-set-foundations/#q-20261014-advance-20261031-r01-at-most-countable-set-0040) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0030-15815 · q-20261014-advance-20261031-r01-at-most-countable-set-0040 ## Original question and givens On Z use f(x)=a*x+b, with nonzero slope a=5, intercept b=2, selected left=-15, right=16. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N). - G001 — On Z use f(x)=a*x+b, with nonzero slope a=5, intercept b=2, selected left=-15, right=16. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-at-most-countable-set](https://ocme.evemisslab.com/en/mko/mko-at-most-countable-set/) ## Step-by-step derivation S001 · calculationBefore affine left nonzero-slope product := mul(5, -15) After affine left nonzero-slope product = -75 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore affine original left image := sum(-75, 2) After affine original left image = -73 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore affine right nonzero-slope product := mul(5, 16) After affine right nonzero-slope product = 80 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore affine original right image := sum(80, 2) After affine original right image = 82 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore fixed integer map f(x)=(5)*x+(2);nonzero slope=5 After for ANY integers x,y:f(x)=f(y) implies(5)*(x-y)=0,hence x=y Conditions: none Subtract the same intercept;the nonzero integer slope times(x−y) is zero.Cancellation in the integers gives x=y for every pair.The displayed left/right check is not the source of universal injectivity. Step identifiers and dependenciesRule: affine_universal_injection_by_integer_cancellation · Uses: G001, S004 S006 · impliesBefore f maps Z injectively to Z;selected images=[-73, 82];selected-pair implication=Yes After encode integer z into N_0 by E(z)=2*z if z>=0,else -2*z-1;even/odd branches disjoint and each injective;E composed with f is injective Conditions: none The two branches yield nonnegative even and odd values,so cross-branch collisions are impossible;within each branch cancel±2.Composition gives an injection into the naturals.Negative slopes and equal selected inputs are allowed. Step identifiers and dependenciesRule: integer_to_natural_encoding_parity_and_universal_composition · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute affine original right image with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute affine right nonzero-slope product with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute affine original left image with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Recompute affine left nonzero-slope product with exact rational arithmetic, not rounded decimals. ## Result (5, 2, -15, 16, -73, 82, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: d7e73de0fef331fd8fc67075b46b7e3f104917a8cc31227472360189adc410b9 - candidate record SHA-256: 4cff69a47880011ee2622f1f66e0abe2d5b3f373fb8c3cc3acd38e791d2f4501 - semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc - question-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0040.json)