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If p(x) denotes the set of all polynomials one indeterminates x over field F, then show that p(x) is a vector space over F with addition defined as addition of polynomials and scalar multiplication defined as the product of polynomials by an element of F. i.e if p(x)={p(x)/p(x)=a₀+a₁x+...+aₙxₙ...}={∑∞,ₙ₌∞ aₙxⁿ for as ∈ f}.

Define addition and scalar multiplication to prove.

Let V be set of real valued continuous function defined as [0,1] such that f(0/3)=2. Show that V is not a vector space over R (reals) under addition and scalar multiplication defined as : (f+g)(x)=f(x)+g(x) for all f,g € V.

(alpha f)(x)= alpha f(x) for all alpha € R, f€V.

What's is mathematics

Solve the problem with 3 different Voting Method.

The Plurality Method with Elimination

The Borda Count Method

The owner of a restaurant decides to poll regular customers to choose which dish she’ll submit to an annual citywide competition. The choices are lemon-crusted salmon (L), crab-stuffed chicken (C), garlic prime rib (G), and wasabi rolls (W). The results are shown in the preference table.

# of votes 8 8 5 5 4

C 1 4 3 2 2

G 3 3 2 1 4

L 4 1 4 4 3

W 2 2 1 3 1