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Question 15804: A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0030-15804 · q-20261014-advance-20261031-r01-at-most-countable-set-0029

Original question and givens

On Z use f(x)=a*x+b, with nonzero slope a=4, intercept b=-9, selected left=-4, right=5. 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=4, intercept b=-9, selected left=-4, right=5. 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.

Step-by-step derivation

S001 · calculation

Before

affine left nonzero-slope product := mul(4, -4)

After

affine left nonzero-slope product = -16

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001

S002 · calculation

Before

affine original left image := sum(-16, -9)

After

affine original left image = -25

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

affine right nonzero-slope product := mul(4, 5)

After

affine right nonzero-slope product = 20

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

affine original right image := sum(20, -9)

After

affine original right image = 11

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

fixed integer map f(x)=(4)*x+(-9);nonzero slope=4

After

for ANY integers x,y:f(x)=f(y) implies(4)*(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 dependencies

Rule: affine_universal_injection_by_integer_cancellation · Uses: G001, S004

S006 · implies

Before

f maps Z injectively to Z;selected images=[-25, 11];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 dependencies

Rule: 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

(4, -9, -4, 5, -25, 11, 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: 0de1ebaa702cb6c2fb0c15c3379a3dcb8c8b3928a4ea9b38dc26fd4aacaed06b
  • candidate record SHA-256: 0093e57b496570bde88af4107d87ca29ade1c3bd677fdd843f81951e1b906307
  • semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc
  • question-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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