Fix non-empty sets A,B with #(A) = m and #(B) = n . How many constant functions f: A->B are there?
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Fix non-empty sets A,B with #(A) = m and #(B) = n . How many constant...
Problem 2. (Stirling number of the second kind.) (1) If A and B are non-empty finite sets, with |A| = m,|B| = n, then how many onto functions are there from A to B? (Hint: Use inclusion-exclusion, and you do not need to simplify the sum.) (2) Suppose m > n. What is the number of ways to distribute m distinct objects into n identical (unordered) containers, with no container left empty? (Derive from (1), and you do not need...
1. A non-empty heap has n nodes. How many interior modes does it have? 2. A non-empty tree has n > 1 nodes. How many of them are interior nodes? 3. A non-empty heap has L leaves. How many nodes does it have?
1. Let A, B be two non-empty sets and f: A + B a function. We say that f satisfies the o-property if VC+0.Vg, h: C + A, fog=foh=g=h. Prove that f is injective if and only if f satisfies the o-property.
Let A and B be two non-empty bounded sets, and A and B are disjoint. Is sup(A U B) = sup(A) + sup(B)? Prove if true, and give a counter example if not.
4. Let A, B CR be non-empty open sets. Prove that AU B is an open set.
State True or False. Type T or F (No other character) Upper case (Assume all these sets are non- empty A-(B-A) U (AB) (AB UACBUABC (AUBUC) If A is a subset of B, then A UB B
State True or False. Type T or F (No other character) Upper case (Assume all these sets are non- empty A-(B-A) U (AB) (AB UACBUABC (AUBUC) If A is a subset of B, then A UB B
Let S ⊂ R be a non-empty set. For any functions f and g from S into R, define d(f,g) := sup{|f(x)−g(x)| : x∈S}. Is d always a metric on the set F of functions from S into R? Why or why not? What does your answer suggest that we do to find a (useful) subset of functions from S to R on which d is a metric, if F does not work? Give a brief justification for your fix.
6. Let A and B be some finite sets with N elements. • Prove that any onto function : A B is an one-to-one function. • Prove that any one-to-one function /: A B is an onto function. • How many different one-to-one functions f: A+B are there?
How many of the following sets of quantum numbers n,l,m I are allowed for the hydrogen atom? (0) 1,0,0 (ii) 1,0,1 (iii) 1,1,0 (iv) 1,1,1 Select one: a. 4 b. none C. 3 d. 2 e. 1
For many real life data sets the decision boundaries are not linear. How is this non-linearity dealt with by SVMs?