This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20260919-integer-foundations · 2026-09-19 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Does 0.5 belong to the integer set Z?
No
0.5 is not an integer; decimal notation must not be classified by appearance alone.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does -1 belong to the integer set Z?
Yes
-1 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 2 belong to the integer set Z?
Yes
2 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does -3 belong to the integer set Z?
Yes
-3 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 4 belong to the integer set Z?
Yes
4 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 5.5 belong to the integer set Z?
No
5.5 is not an integer; decimal notation must not be classified by appearance alone.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 6 belong to the integer set Z?
Yes
6 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does -7 belong to the integer set Z?
Yes
-7 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 8 belong to the integer set Z?
Yes
8 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does -9 belong to the integer set Z?
Yes
-9 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 10.5 belong to the integer set Z?
No
10.5 is not an integer; decimal notation must not be classified by appearance alone.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does -11 belong to the integer set Z?
Yes
-11 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 12 belong to the integer set Z?
Yes
12 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does -13 belong to the integer set Z?
Yes
-13 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 14 belong to the integer set Z?
Yes
14 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 15.5 belong to the integer set Z?
No
15.5 is not an integer; decimal notation must not be classified by appearance alone.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 16 belong to the integer set Z?
Yes
16 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does -17 belong to the integer set Z?
Yes
-17 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 18 belong to the integer set Z?
Yes
18 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does -19 belong to the integer set Z?
Yes
-19 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 20.5 belong to the integer set Z?
No
20.5 is not an integer; decimal notation must not be classified by appearance alone.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does -21 belong to the integer set Z?
Yes
-21 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 22 belong to the integer set Z?
Yes
22 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does -23 belong to the integer set Z?
Yes
-23 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Does 24 belong to the integer set Z?
Yes
24 is a positive integer, zero, or a negative integer, so it belongs to Z.
Reviewed step-by-step derivation
Integer · qf-integer-membership · candidate
Relative to 0, is the integer 0 positive, zero, or negative?
Zero
Relative to 0, 0 is zero. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer -1 positive, zero, or negative?
Negative
Relative to 0, -1 is negative. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer 2 positive, zero, or negative?
Positive
Relative to 0, 2 is positive. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer -3 positive, zero, or negative?
Negative
Relative to 0, -3 is negative. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer 4 positive, zero, or negative?
Positive
Relative to 0, 4 is positive. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer -5 positive, zero, or negative?
Negative
Relative to 0, -5 is negative. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer 6 positive, zero, or negative?
Positive
Relative to 0, 6 is positive. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer -7 positive, zero, or negative?
Negative
Relative to 0, -7 is negative. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer 8 positive, zero, or negative?
Positive
Relative to 0, 8 is positive. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer -9 positive, zero, or negative?
Negative
Relative to 0, -9 is negative. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer 10 positive, zero, or negative?
Positive
Relative to 0, 10 is positive. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer -11 positive, zero, or negative?
Negative
Relative to 0, -11 is negative. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer 12 positive, zero, or negative?
Positive
Relative to 0, 12 is positive. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer -13 positive, zero, or negative?
Negative
Relative to 0, -13 is negative. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer 14 positive, zero, or negative?
Positive
Relative to 0, 14 is positive. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer -15 positive, zero, or negative?
Negative
Relative to 0, -15 is negative. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer 16 positive, zero, or negative?
Positive
Relative to 0, 16 is positive. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer -17 positive, zero, or negative?
Negative
Relative to 0, -17 is negative. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer 18 positive, zero, or negative?
Positive
Relative to 0, 18 is positive. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer -19 positive, zero, or negative?
Negative
Relative to 0, -19 is negative. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer 20 positive, zero, or negative?
Positive
Relative to 0, 20 is positive. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer -21 positive, zero, or negative?
Negative
Relative to 0, -21 is negative. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer 22 positive, zero, or negative?
Positive
Relative to 0, 22 is positive. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer -23 positive, zero, or negative?
Negative
Relative to 0, -23 is negative. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
Relative to 0, is the integer 24 positive, zero, or negative?
Positive
Relative to 0, 24 is positive. Zero is neither positive nor negative.
Reviewed step-by-step derivation
Integer · qf-integer-sign-classification · candidate
What is the additive opposite of 0?
0
0+0=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of -1?
1
-1+1=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of 2?
-2
2+-2=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of -3?
3
-3+3=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of 4?
-4
4+-4=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of -5?
5
-5+5=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of 6?
-6
6+-6=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of -7?
7
-7+7=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of 8?
-8
8+-8=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of -9?
9
-9+9=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of 10?
-10
10+-10=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of -11?
11
-11+11=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of 12?
-12
12+-12=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of -13?
13
-13+13=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of 14?
-14
14+-14=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of -15?
15
-15+15=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of 16?
-16
16+-16=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of -17?
17
-17+17=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of 18?
-18
18+-18=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of -19?
19
-19+19=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of 20?
-20
20+-20=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of -21?
21
-21+21=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of 22?
-22
22+-22=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of -23?
23
-23+23=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
What is the additive opposite of 24?
-24
24+-24=0. This is the additive opposite, not a reciprocal or absolute value.
Reviewed step-by-step derivation
Integer · qf-integer-opposite · candidate
The zero point is the entrance, the positive direction is to the right, and the unit is metres. What signed integer displacement from zero is represented by position 0?
0
The zero point and positive direction are explicit, so 0 metres directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is upward, and the unit is floors. What signed integer displacement from zero is represented by position -1?
-1
The zero point and positive direction are explicit, so -1 floors directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is to the right, and the unit is metres. What signed integer displacement from zero is represented by position 2?
2
The zero point and positive direction are explicit, so 2 metres directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is upward, and the unit is floors. What signed integer displacement from zero is represented by position -3?
-3
The zero point and positive direction are explicit, so -3 floors directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is to the right, and the unit is metres. What signed integer displacement from zero is represented by position 4?
4
The zero point and positive direction are explicit, so 4 metres directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is upward, and the unit is floors. What signed integer displacement from zero is represented by position -5?
-5
The zero point and positive direction are explicit, so -5 floors directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is to the right, and the unit is metres. What signed integer displacement from zero is represented by position 6?
6
The zero point and positive direction are explicit, so 6 metres directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is upward, and the unit is floors. What signed integer displacement from zero is represented by position -7?
-7
The zero point and positive direction are explicit, so -7 floors directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is to the right, and the unit is metres. What signed integer displacement from zero is represented by position 8?
8
The zero point and positive direction are explicit, so 8 metres directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is upward, and the unit is floors. What signed integer displacement from zero is represented by position -9?
-9
The zero point and positive direction are explicit, so -9 floors directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is to the right, and the unit is metres. What signed integer displacement from zero is represented by position 10?
10
The zero point and positive direction are explicit, so 10 metres directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is upward, and the unit is floors. What signed integer displacement from zero is represented by position -11?
-11
The zero point and positive direction are explicit, so -11 floors directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is to the right, and the unit is metres. What signed integer displacement from zero is represented by position 12?
12
The zero point and positive direction are explicit, so 12 metres directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is upward, and the unit is floors. What signed integer displacement from zero is represented by position -13?
-13
The zero point and positive direction are explicit, so -13 floors directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is to the right, and the unit is metres. What signed integer displacement from zero is represented by position 14?
14
The zero point and positive direction are explicit, so 14 metres directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is upward, and the unit is floors. What signed integer displacement from zero is represented by position -15?
-15
The zero point and positive direction are explicit, so -15 floors directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is to the right, and the unit is metres. What signed integer displacement from zero is represented by position 16?
16
The zero point and positive direction are explicit, so 16 metres directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is upward, and the unit is floors. What signed integer displacement from zero is represented by position -17?
-17
The zero point and positive direction are explicit, so -17 floors directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is to the right, and the unit is metres. What signed integer displacement from zero is represented by position 18?
18
The zero point and positive direction are explicit, so 18 metres directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is upward, and the unit is floors. What signed integer displacement from zero is represented by position -19?
-19
The zero point and positive direction are explicit, so -19 floors directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is to the right, and the unit is metres. What signed integer displacement from zero is represented by position 20?
20
The zero point and positive direction are explicit, so 20 metres directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is upward, and the unit is floors. What signed integer displacement from zero is represented by position -21?
-21
The zero point and positive direction are explicit, so -21 floors directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is to the right, and the unit is metres. What signed integer displacement from zero is represented by position 22?
22
The zero point and positive direction are explicit, so 22 metres directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is upward, and the unit is floors. What signed integer displacement from zero is represented by position -23?
-23
The zero point and positive direction are explicit, so -23 floors directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
The zero point is the entrance, the positive direction is to the right, and the unit is metres. What signed integer displacement from zero is represented by position 24?
24
The zero point and positive direction are explicit, so 24 metres directly represents the signed displacement. No unit conversion is introduced.
Reviewed step-by-step derivation
Integer · qf-signed-context-integer · candidate
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