Corticol superficial siderosis

1) A and B being a finite set with |A| > |B| and f : A -> B f being a linear mapping. Show that f is not
injective
2) ? ∈ ℕ

  • being a positive natural Number. A and B being a finite set with |A| > k |B| and f : A -> B f
    being a linear mapping. Prove that there is a ? ∈ ? with |?
    −1
    ({?})| ≥ ? +1
    ?
    −1
    : P(B) -> P(A) being the inverse image of f.
    3) A being a non empty, finite set. An image f called 2-Colouring of a, if you assign a colour to every
    element from A. Show through complete induction that there are 2
    |?|
    variations of 2-Colouring from
    A.
    4) A Word with the length ? ∈ ℕ from the Alphabet ∑ := {0, 1} is a
    Squence w = (?0
    , ?1
    , … , ??−1) ∈ ∑
    ?
    . In the case ? =0 it’s the empty word/sequence(). ? ∈ ℕ .
    Prove that there are 2
    ?+1 − 1 Words with a length of maximum n over the Alphabet ∑
    5) ? ∈ ℕ, A a finite set with |A|=n and ∑ := {0, 1}. F being the amount of all 2-Colouring from A, and W
    the amount of all Words of the length ? over the Alphabet ∑.
    Show that there is a Bijection between F and W.
    6) G is a connected graph and ? ∈ ℕ is chosen that for all blocks ? ⊆ G applies |?(?)| ≤ ?.
    a) State an algorithm (without reasoning) that provides a largest (concerning Cardinality)
    independent set in G and it’s runtime is in ?(?(?)?(|?(?)|)).
    f being a function described by f: ℕ → ℕ that only depends on k while p: ℕ → ℕ is a
    polynomial that only depends on |?(?)|.
    b) Prove the correctness of the algorithm from a
    c) Justify why the runtime from a is in ?(?(?)?(|?(?)|)).
    7) ? ∈ ℕ
  • a positive natural number. We know one Graph G, that is able to be made c-planar, that if
    there is a set ? ⊆ V(G) with maximum of c nodes that: G – S planar.
    a) Give an example graph G with a maximum of 10 nodes that can’t be made 2-planar but 3-planar.
    Show that your example is correct by selecting a set ? ⊆ V(G) of Cardinality 3 so that G-S
    planar.
    b) G is a Graph that can be made 5-planar. How many pairwise node-disjoint copies of ?5 can G
    contain at a maximum. Provide the best boundaries and reasons for your answer.
    c) G is a Graph with ? ∈ ℕ
  • blocks, all blocks are able to be made into c-planars. How many pairwise
    node-disjoint copies of ?5 can G contain at a maximum? State the best boundaries in dependency
    to b and c plus your reasoning behind it.
    d) G being a graph and B a block from G, prove ?
    (?) = ????⊆G?
    (?)
    8) X is a finite set with n:= |?|. A set ? ⊆ ?(?) is a Clutter, if for all ?,? ∈ ? applies: If ? ⊆ ? then A=B
    a) X:={1,2,3,4}. State a Clutter ? ⊆ ?(?) with |?| = (
    4
    2
    )
    b) Clutter ? ⊆ ?(?). Show that |?| ≤ (
    ?
    |
    ?
    2
    |
    )
    9) X is a finite set with n:= |?|. A set ? ⊆ ?(?) is a Clutter.
    F is simple, if for all ?, ? ∈ ? with ? ≠ ? applies: |? ∩ ?| ≤ 1 and |?| ≥ 2.
    F is forestlike, if for all sub-families ? ⊆ ? an ? ∈ ? and an element ? ∈ ? exists, that ? ∈ ?
    with ? ≠ ? applies: if ? ∩ ? ≠ 0 then ? ∩ ? = {?}.
    C ⊆ ?(?) is a set that is a subset of X. A real 2-colouring of C is a transformation f:X->{red,blue},
    so that ? ∈ ? applies ? ∩ ?
    −1
    ({???}) ≠ 0 and ? ∩ ?
    −1
    ({????}) ≠ 0.
    a) G is a connected graph with a minimum of 2 nodes. We define the following set:
    ?(?) ≔ {?(?)|? ⊆ ? and ? is a block from ?} ⊆ ?(?(?))
    Prove that B(G) is a simple and forestlike Clutter.
    b) Show that there is a real 2-colouring for every simple and forestlike Clutter ? ⊆ ?(?).

IS IT YOUR FIRST TIME HERE? WELCOME

USE COUPON "11OFF" AND GET 11% OFF YOUR ORDERS