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Question 1151: Complete Zero Set of an Affine Function

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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R0001-01151 · q-20261003-r05-linear-function-0051

Original question and givens

Over R, f(x)=(0)x+(0). Find its complete zero set, using R, ∅, or a set of reduced rational numbers.

  • G001 — Over R, f(x)=(0)x+(0). Find its complete zero set, using R, ∅, or a set of reduced rational numbers.

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 · implies

Before

f:R→R; f(x)=mx+b

After

m=0 gives a constant affine function

Conditions: none

An affine function allows m=0; do not impose the nondegenerate-equation condition on every affine function.

Step identifiers and dependencies

Rule: affine_domain · Uses: G001

S002 · implies

Before

0x+(0)=0

After

m≠0: No

Conditions: none

Check whether division by m is allowed; classify constant functions separately.

Step identifiers and dependencies

Rule: zero_set_case · Uses: G001, S001

S003 · implies

Before

0x+(0)=0

After

constant equation true: Yes

Conditions: none

If the constant is0 every real is a zero; otherwise there is no zero.

Step identifiers and dependencies

Rule: constant_zero_equation · Uses: G001, S002

S004 · implies

Before

constant equation independent of x

After

zero set=R

Conditions: none

This concerns all reals or the empty set, not just sampled inputs.

Step identifiers and dependencies

Rule: complete_real_zero_set · Uses: G001, S003

Reverse checks

  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

R

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: 548eb0ee8e1d30c7518251d561cf185db6ff334208f5976d048fd0163b105195
  • candidate record SHA-256: e8cc50a2b10cd5dfde128cc4c50b0268f241e0bc4acbcc3932a9020164e5d4b9
  • semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc
  • question-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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