19 Matching Annotations
  1. Jul 2020
    1. Let N = [S A B] and T = [0 1]. Find the language generated by each grammar [N T S P] when the set P of productions

      As a reminder:

      • N = Nonterminals
      • T = Terminals
      • S = Start
      • P = Productions/Rules
    2. Eulerian

      Here's a good resource.

    3. Pseudographs

      These are often used in state machines. A state will keep looping to itself until it receives a trigger to pass to the next state.

  2. Jun 2020
    1. Start the branching with 4 powers of 2: 20, 21, 22 and 23

      We branch with these powers because the prime factorization has \(2^3\). If it were instead \(2^5\), we would branch with \(2^0\), \(2^1\), \(2^2\), \(2^3\), \(2^4\), and \(2^5\), and we would have a branching factor of 6.

    2. 23⋅32⋅5123⋅32⋅512^3 \cdot 3^2 \cdot 5^1, are found at the bottom of this tree

      Since this seems to render rather messily on some browsers, what this sentence means to say is:

      "All factors of 360, whose prime factorization is \(2^3\), \(3^2\), \(5^1\), are found at the bottom of this tree."

  3. May 2020
    1. ≡ 8931 ⋅ 89364 ⋅ 893128 ⋅ 893512 ⋅ 8932048

      Consult the previous table. Replace each 893^n with the corresponding CM4721.

    2. mod 7 MMI of 3

      7 % 3 = 1

    3. CSF

      "[A number is in] complete standard form (CSF) if its factors are listed as powers (including 0 and 1)."

    4. limit

      "limit" represents how many primes to search for. Try estimating how many primes are between 0 and p100rle.

    5. ELM

      Since you want specifically the permutations with ELM, you can treat ELM as a single unit.

    6. Concretize

      "Concretize" is supposed to mean the opposite of "make abstract."

      https://www.google.com/search?q=concretize

    7. onto

      DEI: "a function whose range is the same as its codomain (that is, an onto function produces (for the right inputs) all possible outputs)."

    8. one-to-one

      DEI: "a function each of whose outputs is generated by only one input."

    9. functions

      "Functions" just means ways the elements can be combined.

      From DEI: "A function is just an object that takes objects and gives other objects."

    10. Let A0, A1, A2, …, An be subsets of a set S. A partition of S is formed by these subsets iff they are nonempty, disjoint and exhaust S. In symbols: Ai ≠ ∅. Ai ∩ Aj = ∅ if i ≠ j. ⋃⋃\bigcup Ai = S. If these three conditions hold, we say these Ai subsets partition S.

      A group of subsets is a partition of a superset if none of them are empty, they do not share any members, and all the subsets combined equal the superset.

    11. R is antisymmetric iff ∀∀\forall x ∀∀\forall y [xRy ∧ yRx → x = y].

      For any x and y in a set, if x is related to y and y is related to x, for it to be antisymmetric x must equal y.

      Example:

      We have the set [1 2 3].

      For a relation to be antisymmetric, if we have (1 2) we must not have (2 1). In other words, if there is any symmetry it must be between the same elements, such as (1 1).

      Additionally, if we do not have any element related to any element it is vacuously antisymmetric.

    12. R is reflexive iff ∀∀\forall x [x ∈∈\in U → xRx]

      For each x in the set, x must be related to x for the relation to be reflexive. In other words, each element must be related to itself.

      Example:

      We have the set [1 2 3].

      For a relation to be reflexive, it must have (1 1), (2 2), and (3 3).

    13. R is symmetric iff ∀∀\forall x ∀∀\forall y [xRy → yRx].

      For any x and y in a set, if x is related to y then y must be related to x for the relation to be symmetric.

      Example:

      We have the set [1 2 3].

      For a relation to be symmetric, if you have (1 2) you must have (2 1). Equally, (1 1) is symmetric.

    14. R is transitive iff ∀∀\forall x ∀∀\forall y ∀∀\forall z [xRy ∧ yRz → xRz].

      For any x, y, and z in a set, if x is related to y and y is related to z, then z must also be related to z for the relation to be transitive.

      Example:

      We have the set [1 2 3].

      For a relation to be transitive, if we have (1 2) and (2 3), we must have (1 3). Likewise, any reflexive pair is transitive.

      Additionally, if we do not have xRy and yRz (even if x, y, or z are the same element), the relation is vacuously transitive.