{
  "index": "1964-B-2",
  "type": "COMB",
  "tag": [
    "COMB"
  ],
  "difficulty": "",
  "question": "2. Let \\( S \\) be a set of \\( n>0 \\) elements, and let \\( A_{1}, A_{2}, \\ldots, A_{k} \\) be a family of distinct subsets, with the property that any two of these subsets meet. Assume that no other subset of \\( S \\) meets all of the \\( A_{i} \\).\n\nProve that \\( k=2^{n-1} \\).",
  "solution": "Solution. There are \\( 2^{n} \\) distinct subsets of \\( S \\) which we may consider as being arranged in \\( 2^{n-1} \\) complementary pairs. If \\( k \\), the number of subsets in the family, is greater than \\( 2^{n-1} \\), then two of these subsets would be complementary, and these two subsets would not intersect. On the other hand, if \\( k<2^{n-1} \\), we can find two complementary sets \\( X \\) and \\( Y \\), neither of which is among the given sets \\( A_{1}, A_{2}, \\ldots, A_{k} \\). But no set not among the \\( A \\) 's meets all the \\( A \\) 's, so there is a set \\( A_{i} \\) with \\( A_{i} \\cap X=\\emptyset \\), i.e., \\( A_{i} \\subseteq Y \\). Similarly there is a set \\( A_{j} \\) with \\( A_{i} \\subseteq X \\). But then \\( A_{i} \\cap A_{i} \\subseteq X \\cap Y=\\emptyset \\), which contradicts the fact that any two of the \\( A \\) 's meet. So the only possibility is \\( k=2^{n-1} \\).\n\nRemark. Evidently, such a family is given by the collection of all subsets of \\( S \\) which contain a fixed element of \\( S \\). There are, however, other possibilities; for example, if \\( S=\\{1,2,3\\} \\), then \\( S,\\{1,2\\},\\{1,3\\},\\{2,3\\} \\) is such a family.",
  "vars": [
    "S",
    "A_1",
    "A_2",
    "A_k",
    "A_i",
    "A_j",
    "X",
    "Y"
  ],
  "params": [
    "n",
    "k"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "S": "setbase",
        "A_1": "alphaone",
        "A_2": "alphatwo",
        "A_k": "alphakey",
        "A_i": "alphaitem",
        "A_j": "alphajay",
        "X": "setexes",
        "Y": "setwhyy",
        "n": "elemsize",
        "k": "familysize"
      },
      "question": "2. Let \\( setbase \\) be a set of \\( elemsize>0 \\) elements, and let \\( alphaone, alphatwo, \\ldots, alphakey \\) be a family of distinct subsets, with the property that any two of these subsets meet. Assume that no other subset of \\( setbase \\) meets all of the \\( alphaitem \\).\n\nProve that \\( familysize=2^{elemsize-1} \\).",
      "solution": "Solution. There are \\( 2^{elemsize} \\) distinct subsets of \\( setbase \\) which we may consider as being arranged in \\( 2^{elemsize-1} \\) complementary pairs. If \\( familysize \\), the number of subsets in the family, is greater than \\( 2^{elemsize-1} \\), then two of these subsets would be complementary, and these two subsets would not intersect. On the other hand, if \\( familysize<2^{elemsize-1} \\), we can find two complementary sets \\( setexes \\) and \\( setwhyy \\), neither of which is among the given sets \\( alphaone, alphatwo, \\ldots, alphakey \\). But no set not among the \\( A \\) 's meets all the \\( A \\) 's, so there is a set \\( alphaitem \\) with \\( alphaitem \\cap setexes=\\emptyset \\), i.e., \\( alphaitem \\subseteq setwhyy \\). Similarly there is a set \\( alphajay \\) with \\( alphaitem \\subseteq setexes \\). But then \\( alphaitem \\cap alphaitem \\subseteq setexes \\cap setwhyy=\\emptyset \\), which contradicts the fact that any two of the \\( A \\) 's meet. So the only possibility is \\( familysize=2^{elemsize-1} \\).\n\nRemark. Evidently, such a family is given by the collection of all subsets of \\( setbase \\) which contain a fixed element of \\( setbase \\). There are, however, other possibilities; for example, if \\( setbase=\\{1,2,3\\} \\), then \\( setbase,\\{1,2\\},\\{1,3\\},\\{2,3\\} \\) is such a family."
    },
    "descriptive_long_confusing": {
      "map": {
        "S": "longitude",
        "A_1": "panorama",
        "A_2": "sandwich",
        "A_k": "accordion",
        "A_i": "lighthouse",
        "A_j": "butterfly",
        "X": "waterfall",
        "Y": "pineapple",
        "n": "teaspoon",
        "k": "harmonica"
      },
      "question": "2. Let \\( longitude \\) be a set of \\( teaspoon>0 \\) elements, and let \\( panorama, sandwich, \\ldots, accordion \\) be a family of distinct subsets, with the property that any two of these subsets meet. Assume that no other subset of \\( longitude \\) meets all of the \\( lighthouse \\).\n\nProve that \\( harmonica=2^{teaspoon-1} \\).",
      "solution": "Solution. There are \\( 2^{teaspoon} \\) distinct subsets of \\( longitude \\) which we may consider as being arranged in \\( 2^{teaspoon-1} \\) complementary pairs. If \\( harmonica \\), the number of subsets in the family, is greater than \\( 2^{teaspoon-1} \\), then two of these subsets would be complementary, and these two subsets would not intersect. On the other hand, if \\( harmonica<2^{teaspoon-1} \\), we can find two complementary sets \\( waterfall \\) and \\( pineapple \\), neither of which is among the given sets \\( panorama, sandwich, \\ldots, accordion \\). But no set not among the \\( A \\) 's meets all the \\( A \\) 's, so there is a set \\( lighthouse \\) with \\( lighthouse \\cap waterfall=\\emptyset \\), i.e., \\( lighthouse \\subseteq pineapple \\). Similarly there is a set \\( butterfly \\) with \\( lighthouse \\subseteq waterfall \\). But then \\( lighthouse \\cap lighthouse \\subseteq waterfall \\cap pineapple=\\emptyset \\), which contradicts the fact that any two of the \\( A \\) 's meet. So the only possibility is \\( harmonica=2^{teaspoon-1} \\).\n\nRemark. Evidently, such a family is given by the collection of all subsets of \\( longitude \\) which contain a fixed element of \\( longitude \\). There are, however, other possibilities; for example, if \\( longitude=\\{1,2,3\\} \\), then \\( longitude,\\{1,2\\},\\{1,3\\},\\{2,3\\} \\) is such a family."
    },
    "descriptive_long_misleading": {
      "map": {
        "S": "sequencebag",
        "A_1": "supersetuno",
        "A_2": "supersetdos",
        "A_k": "supersetmega",
        "A_i": "supersetvar",
        "A_j": "supersetalt",
        "X": "overlapset",
        "Y": "unionset",
        "n": "infinitesize",
        "k": "minicount"
      },
      "question": "2. Let \\( sequencebag \\) be a set of \\( infinitesize>0 \\) elements, and let \\( supersetuno, supersetdos, \\ldots, supersetmega \\) be a family of distinct subsets, with the property that any two of these subsets meet. Assume that no other subset of \\( sequencebag \\) meets all of the \\( supersetvar \\).\n\nProve that \\( minicount=2^{infinitesize-1} \\).",
      "solution": "Solution. There are \\( 2^{infinitesize} \\) distinct subsets of \\( sequencebag \\) which we may consider as being arranged in \\( 2^{infinitesize-1} \\) complementary pairs. If \\( minicount \\), the number of subsets in the family, is greater than \\( 2^{infinitesize-1} \\), then two of these subsets would be complementary, and these two subsets would not intersect. On the other hand, if \\( minicount<2^{infinitesize-1} \\), we can find two complementary sets \\( overlapset \\) and \\( unionset \\), neither of which is among the given sets \\( supersetuno, supersetdos, \\ldots, supersetmega \\). But no set not among the \\( A \\) 's meets all the \\( A \\) 's, so there is a set \\( supersetvar \\) with \\( supersetvar \\cap overlapset=\\emptyset \\), i.e., \\( supersetvar \\subseteq unionset \\). Similarly there is a set \\( supersetalt \\) with \\( supersetvar \\subseteq overlapset \\). But then \\( supersetvar \\cap supersetvar \\subseteq overlapset \\cap unionset=\\emptyset \\), which contradicts the fact that any two of the \\( A \\) 's meet. So the only possibility is \\( minicount=2^{infinitesize-1} \\).\n\nRemark. Evidently, such a family is given by the collection of all subsets of \\( sequencebag \\) which contain a fixed element of \\( sequencebag \\). There are, however, other possibilities; for example, if \\( sequencebag=\\{1,2,3\\} \\), then \\( sequencebag,\\{1,2\\},\\{1,3\\},\\{2,3\\} \\) is such a family."
    },
    "garbled_string": {
      "map": {
        "S": "qzxwvtnp",
        "A_1": "hjgrksla",
        "A_2": "mfldopqe",
        "A_k": "rjsundel",
        "A_i": "vpknaoly",
        "A_j": "bqterwzc",
        "X": "wnepxqiv",
        "Y": "slodmtak",
        "n": "gdrfmaue",
        "k": "fsqnjlop"
      },
      "question": "2. Let \\( qzxwvtnp \\) be a set of \\( gdrfmaue>0 \\) elements, and let \\( hjgrksla, mfldopqe, \\ldots, rjsundel \\) be a family of distinct subsets, with the property that any two of these subsets meet. Assume that no other subset of \\( qzxwvtnp \\) meets all of the \\( vpknaoly \\).\n\nProve that \\( fsqnjlop=2^{gdrfmaue-1} \\).",
      "solution": "Solution. There are \\( 2^{gdrfmaue} \\) distinct subsets of \\( qzxwvtnp \\) which we may consider as being arranged in \\( 2^{gdrfmaue-1} \\) complementary pairs. If \\( fsqnjlop \\), the number of subsets in the family, is greater than \\( 2^{gdrfmaue-1} \\), then two of these subsets would be complementary, and these two subsets would not intersect. On the other hand, if \\( fsqnjlop<2^{gdrfmaue-1} \\), we can find two complementary sets \\( wnepxqiv \\) and \\( slodmtak \\), neither of which is among the given sets \\( hjgrksla, mfldopqe, \\ldots, rjsundel \\). But no set not among the \\( A \\) 's meets all the \\( A \\) 's, so there is a set \\( vpknaoly \\) with \\( vpknaoly \\cap wnepxqiv=\\emptyset \\), i.e., \\( vpknaoly \\subseteq slodmtak \\). Similarly there is a set \\( bqterwzc \\) with \\( vpknaoly \\subseteq wnepxqiv \\). But then \\( vpknaoly \\cap vpknaoly \\subseteq wnepxqiv \\cap slodmtak=\\emptyset \\), which contradicts the fact that any two of the \\( A \\) 's meet. So the only possibility is \\( fsqnjlop=2^{gdrfmaue-1} \\).\n\nRemark. Evidently, such a family is given by the collection of all subsets of \\( qzxwvtnp \\) which contain a fixed element of \\( qzxwvtnp \\). There are, however, other possibilities; for example, if \\( qzxwvtnp=\\{1,2,3\\} \\), then \\( qzxwvtnp,\\{1,2\\},\\{1,3\\},\\{2,3\\} \\) is such a family."
    },
    "kernel_variant": {
      "question": "Let \\(\\Gamma\\) be a finite set with \\(r\\ge 1\\) elements.  Consider distinct subsets \\(B_{1},B_{2},\\dots ,B_{m}\\subseteq \\Gamma\\) such that\n\n(1) every two of them overlap: for all \\(i\\neq j\\) we have \\(B_{i}\\cap B_{j}\\neq\\varnothing\\);\n\n(2) a subset of \\(\\Gamma\\) that meets every \\(B_{i}\\) must already be one of the \\(B_{i}\\).\n\nProve that \\(m=2^{r-1}.\\)",
      "solution": "Because |Γ|=r, Γ has 2^r subsets.  Arrange these subsets in complementary pairs (X, Γ\\X); there are exactly 2^{r-1} such pairs.\n\nCase 1: m>2^{r-1}.  By the pigeon-hole principle two of the family members, say B_p and B_q, must lie in the same complementary pair; hence B_q=Γ\\B_p.  Then B_p∩B_q=∅, contradicting the overlap condition (1).  Therefore m≤2^{r-1}.\n\nCase 2: m<2^{r-1}.  Some complementary pair (X,Γ\\X) is missing from the family.  Because neither member of the pair is a B_i, each of them fails to meet at least one B_i (by property (2)).  Thus there exists an index s with B_s∩X=∅, i.e. B_s⊆Γ\\X, and an index t with B_t∩(Γ\\X)=∅, i.e. B_t⊆X.  But then B_s∩B_t⊆(Γ\\X)∩X=∅, again contradicting (1).  Hence m≥2^{r-1}.\n\nCombining these gives m=2^{r-1}.\n\nRemark.  One way to realize such a family is to fix any single element g∈Γ and take all subsets containing g.  There are 2^{r-1} of them, and they satisfy conditions (1) and (2).",
      "_meta": {
        "core_steps": [
          "Pair the 2^n subsets of S into 2^{n-1} complementary pairs (X , S\\X).",
          "If k > 2^{n-1}, two A_i lie in the same pair and are complementary, hence disjoint – contradicting pairwise intersection.",
          "If k < 2^{n-1}, choose a complementary pair (X , S\\X) missing from the family; maximality forces one A_i ⊆ X^c and another A_j ⊆ X, making them disjoint – contradiction.",
          "Both inequalities are impossible, so k = 2^{n-1}."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Stipulation that the ground set is non-empty",
            "original": "n>0"
          },
          "slot2": {
            "description": "Wording of the pairwise-intersection condition",
            "original": "any two of these subsets meet"
          },
          "slot3": {
            "description": "Choice of symbols for the ground set and the family (S, A_i, k)",
            "original": "S, A_1,…,A_k, k"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}