60 lines
3.4 KiB
YAML
60 lines
3.4 KiB
YAML
dataset_name: abstract_algebra
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description: The following are multiple choice questions (with answers) about abstract
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algebra.
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fewshot_config:
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sampler: first_n
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samples:
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- question: 'Statement 1 | Every element of a group generates a cyclic subgroup of
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the group. Statement 2 | The symmetric group S_10 has 10 elements.
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(A) True, True (B) False, False (C) True, False (D) False, True'
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target: Let's think step by step. A cyclic group is a group that is generated
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by a single element. Hence a subgroup generated by a single element of a group
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is cyclic and Statement 1 is True. The answer is (C).
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- question: 'The symmetric group $S_n$ has $
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actorial{n}$ elements, hence it is not true that $S_{10}$ has 10 elements.
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Find the characteristic of the ring 2Z.
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(A) 0 (B) 3 (C) 12 (D) 30'
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target: Let's think step by step. A characteristic of a ring is R is $n$ if the
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statement $ka = 0$ for all $a\in 2Z$ implies that $k$ is a multiple of $n$.
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Assume that $ka = 0$ for all $a\in 2Z$ for some $k$. In particular $2k = 0$.
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Hence $k=0$ and $n=0$. The answer is (A).
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- question: 'Statement 1| Every function from a finite set onto itself must be one
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to one. Statement 2 | Every subgroup of an abelian group is abelian.
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(A) True, True (B) False, False (C) True, False (D) False, True'
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target: "Let's think step by step. Statement 1 is true. Let $S$ be a finite set.\
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\ If $f:S \nightarrow S$ is a onto function, then $|S| = |f(S)|$. If $f$ was\
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\ not one to one, then for finite domain $S$ the image would have less than\
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\ $S$ elements, a contradiction.\nStatement 2 is true. Let $G$ be an abelian\
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\ group and $H$ be a subgroup of $G$. We need to show that $H$ is abelian. Let\
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\ $a,b \\in H$. Then $a,b \\in G$ and $ab=ba$. Since $G$ is abelian, $ab=ba$.\
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\ Since $H$ is a subgroup of $G$, $ab \\in H$. Therefore, $ab=ba$ and $H$ is\
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\ abelian. The answer is (A)."
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- question: 'Statement 1 | If aH is an element of a factor group, then |aH| divides
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|a|. Statement 2 | If H and K are subgroups of G then HK is a subgroup of G.
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(A) True, True (B) False, False (C) True, False (D) False, True'
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target: Let's think step by step. Statement 2 is false. Let $H$ be a subgroup
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of $S_3$ generated by the cycle $(1,2)$ and $K$ be a subgroup of $S_3$ generated
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by the cycle $(1,3)$. Both $H$ and $K$ have two elements, the generators and
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the identity. However $HK$ contains cycles (1,2), (1,3) and (2,3,1), but the
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inverse of (2,3,1) is (2,1,3) and it does not belong to HK, hence HK is not
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a subgroup. The answer is (B).
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- question: 'Find all c in Z_3 such that Z_3[x]/(x^2 + c) is a field.
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(A) 0 (B) 1 (C) 2 (D) 3'
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target: 'Let''s think step by step. Z_3[x]/(x^2 + c) is a field if and only if
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x^2 + c does not have roots in Z_3. That is x^2 + c != 0 for every x in Z_3.
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If c = 0, then x^2 + c = x^2 has root 0. If c = 1 then x^2 + c = x^2 + 1 = 0
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+ 1 for x = 0, 1 + 1 = 2 for x = 1 and 1 + 1 = 2 for x = 2, hence x^2 + 1 does
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not have any roots. For c = 2 the polynomial x^2 + 2 has two roots at x = 1
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and x = 2. Hence Z_3[x]/(x^2 + c) is a field if and only if c = 1. The answer
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is (B).'
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tag: mmlu_flan_cot_fewshot_stem
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include: _mmlu_flan_cot_fewshot_template_yaml
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task: mmlu_flan_cot_fewshot_abstract_algebra
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