Problem Let T be a set and partially order P(T) by inclusion. Let CCP(P(T)) be a chain and let F be a finite subset of UUEC U Prove that there is some UEC such that FCU. Hint: Clearly this is an induction problem.

4. Using mathematical induction, prove that if A₁, A2,, An are subsets of a universal set U, then UAk=nAk. CS Scanned with CamScanner

Define a one-to-one correspondence f : X → Y. Explain why f is one-to-one andonto.

Give a real-world example of the inclusion/exclusion principle that involves at least two finite sets.

please show the correct answer and steps (12)

Let G := (V ,EG) be a 4-regular tree, i.e. an infinite connected undirected graph without cycles and such that each vertex has exactly 4 neighbors, the picture is a finite part of it. Recall that Th(G) denotes the theory of G, i.e. the set of all sentences that G satisfies. Using the Compactness theorem, prove that Th(G) has a disconnected model.  Hint: Add two new constant symbols and sneakily make them farther and farther from each other.

Let A = {1, 2, 3},         B = {{1}, 2, {1, 3}},         C = {1, {1}, 3}.   Compute the following sets:                                                                         2A∩C − B,         (A ∩ B) × (A − C),         (B ∪ C) − A.

Show that if A and B are sets with A ⊆ B then A ∪ B = B. (Hint: Show that x ∈ A ∪ B implies x ∈ B and vice versa.)

Exercise c. Strict matchingConsider the following generalization of the maximum matching problem, which we callStrict-Matching. Recall that a matching in an undirected graph G = (V, E) is a setof edges such that no distinct pair of edges {a, b} and {c, d} have endpoints that areequal: {a, b} ∩ {c, d} = ∅. Say that a strict matching is matching with the propertythat no pair of distinct edges have endpoints that are connected by an edge: {a, c} ̸∈ E,{a, d} ̸∈ E, {b, c} ̸∈ E, and {b, d} ̸∈ E. (Since a strict matching is also a matching, wealso require {a, b} ∩ {c, d} = ∅.) The problem Strict-Matching is then given a graphG and an integer k, does G contain a strict matching with at least k edges.Prove that Strict-Matching is NP-complete.

On S = f2; 4; 6; 10; 12; 20; 30; 60g we work with the divisibility relation R:xRy , x divides y:(i) Draw the Hasse diagram of R.(ii) Find all R-minimal elements.(iii) Find all R-maximal elements.(iv) Find all non-empty subsets U  S, so that 6 is an R-lowerbound of U.(v) Let A = f6; 10; 12g. If A admits an R-supremum,  nd it. If A admits an R-in mum,  nd it.(vi) Is R a lattice?