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Suppose that C = {f(x) : f is a realvalued continuous function on [0, 1]} . Dene addition 6 on C by the unusual rule U$8)(x) E max (f(x),g(x)), for g E C and x E [0,1]. (f 69 9W) is n u u as u l u This 6 addition comes from the maxplus algebra and is of practical use in control theory. Define scalar multiplication by the usual rule (atf)(x) aim), for f e C, A e R and x 6 [0,1] . Which of the following 10 properties of a vector space are satised? LCIosure under Addition. Iff, g E C , thenf 6 g E C. 2.Associative Law ofAddition. lff, g, h E C , then (f 6 g) 6 h =f 6 (g 6 h). 3. Commutative Law of Addition. Iff, g E C , thenf 6 g = 3 6f. 4. Existence of Zero. There exists an element 0 E C such that, for allf E C ,f 63 0 =f. 5. Existence of Negative. For each f E C there exist an element h E C (usually written as f ) such that f 6 h = 0. 5. Closure under Multiplication by a Scalar. lff E C and A E IR , then if E C. 7.Associative Law of Multiplication by a Scalar. If at, ,u E IR and f E C then 1104f) = (my. a. Scalar Multiplication Identity. If f E C. then 1)\" = f. 9. Scalar Distributive Law. ILL}; E IR andf E C , then (A +)f = if 6p)\". 10. Vector Distributive Law. lfl E IR andf, g E C . then 3(f 6 g) = 11f 6 lg. Enter your answer as a subset of {1 ,2,3,...,10}. {12,345,310} *0 Q B