25 lines
1.1 KiB
JSON
25 lines
1.1 KiB
JSON
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[
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{
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"problem": "For any natural number n, 0 + n = n.",
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"level": "SF foundations level 1",
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"type": "Logical Foundations",
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"solution": [
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"Consider some natural number n. We want to show 0 + n = n. ",
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"By using definition of addition on both sides, LHS and RHS are now equal, done."
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]
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},
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{
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"problem": "For any natural number n, n + 0 = n.",
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"level": "SF foundations level 1",
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"type": "Logical Foundations",
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"solution": [
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"Consider some natural number n. The proof will be by induction. ",
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"The base case n=0, so we have 0 + 0 = 0, which holds by the definition of addion. ",
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"Consider the inductive case, so we want to show (k + 1) + 0 = (k + 1) for any k < n assuming the IH holds for such k (IH: k + 0 = k). ",
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"By the IH we have (k + 1) + 0 = (k + 1). ",
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"By def of addition we have (k + 0) + 1 = (k + 1). ",
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"By the induction hypothesis (IH) we have k + 0 = k. ",
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"LHS and RHS are equal so proof is complete."
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]
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}
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]
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