answer:The problem states that we have a binary operation `*` defined on the set of integers mathbb{Z} which satisfies the following axioms: (a) x * (x * y) = y for all x, y in mathbb{Z}, (b) (x * y) * y = x for all x, y in mathbb{Z}. We need to analyze whether the following claims by the professor are true or false based on the given axioms: (C1) The operation `*` is commutative: x * y = y * x for all x, y in mathbb{Z}. (C2) The operation `*` is associative: (x * y) * z = x * (y * z) for all x, y, z in mathbb{Z}. Let's go through each claim step-by-step. Analyzing (C1) Commutativity: We need to determine whether x * y = y * x holds. To do so, we start by formulating the given axioms using a new notation, as follows: 1. If (x * y) = z, then replace x * y with z in the axioms: (a) x * z = y, i.e., x * (x * y) = y becomes x * (x * y) = y (already in this form). (b) z * y = x, i.e., (x * y) * y = x becomes (x * y) * y = x (already in this form). Given: (x * y) = z implies x * (x * y) = y. To prove: x * y = y * x. Let's follow the sequence of transformations based on the given axioms: 1. Suppose (x * y) = z, so we have both x * (x * y) = y and (x * y) * y = x. 2. Therefore, (x * y) * y = x, and reassign each to (z * y = x). 3. Next, apply (x * (x * y) = y) and reassign each to (x * z = y). Using both transformations: - (x * y) = z rightarrow z = (x * y), - (z * x) = y rightarrow (y * x) = z. Given (x * y) rightarrow (x * y = z) rightarrow (z * x) = y rightarrow (y * x) = z. Now conclude: x * y = (z equiv y * x). Thus by (x * y = y * x), the operation `*` is confirmed to be commutative. Analyzing (C2) Associativity: We need to check whether (x * y) * z = x * (y * z) holds. # Counterexample for (C2): Let's take a specific example: Define x * y = -(x + y). Validate the claims: - (x * y) = -(x + y). Now evaluate using specific integers scenarios: - (x = 1, y = 2, z = 3): - (1 * 2) * 3 = (-(1 + 2)) * 3 = -3 * 3, - 1 * (2 * 3) = 1 * (-(2 + 3)) = 1 * -5 = -(1 - 5). In this proof, observe: - (x * y) = (-3 * 3) = -6 - 1 * (-5) = -6 proves not generally (x * y) * z = x * ( y * z) meaning **not** associative. Concludingly to axioms, the following claims: mathbf{C1 : True ; C2 : False}. # Conclusion: [ boxed{text{(C1) is true; (C2) is false.}} ]

answer:((1)) Let (B(x_{1},y_{1})) and (C(x_{2},y_{2})), When the slope of line (l) is (dfrac{1}{2}), the equation of (l) is (y=dfrac{1}{2}(x+4)), i.e., (x=2y-4), From (begin{cases} x^{2}=2py x=2y-4 end{cases}), we get (2y^{2}-(8+p)y+8=0), thus (begin{cases} y_{1}y_{2}=4 y_{1}+y_{2}=dfrac{8+p}{2} end{cases}), Also, since (overrightarrow{AC}=4overrightarrow{AB}), we have (y_{2}=4y_{1}), From these three expressions and (p > 0), we get (y_{1}=1), (y_{2}=4), (p=2), Therefore, the equation of the parabola is (x^{2}=4y). Thus, the answer for part ((1)) is boxed{x^{2}=4y}. ((2)) Let (l: y=k(x+4)), and the midpoint of (BC) be ((x_{0},y_{0})) From (begin{cases} x^{2}=4y y=k(x+4) end{cases}), we get (x^{2}-4kx-16k=0), thus (x_{0}=2k, y_{0}=k(x_{0}+4)=2k^{2}+4k), The equation of the perpendicular bisector is (y-2k^{2}-4k=-dfrac{1}{k}(x-2k)), Therefore, the (y)-intercept of the perpendicular bisector of segment (BC) is: (b=2k^{2}+4k+2=2(k+1)^{2}), From (Delta =16k^{2}+64k > 0), we get (k > 0) or (k < -4), Therefore, the range of (b) is boxed{(2,+infty)}.

answer:Let's denote Tim's income as T, Mart's income as M, and Juan's income as J. According to the information given: 1) Mart's income is 60 percent more than Tim's income: M = T + 0.60T M = 1.60T 2) Mart's income is 64 percent of Juan's income: M = 0.64J From the two equations above, we can equate the expressions for M: 1.60T = 0.64J Now, we want to find out what percentage less Tim's income is than Juan's income. To do this, we need to express T as a percentage of J. First, let's solve for T in terms of J using the equation we have: T = (0.64J) / 1.60 T = 0.40J This means that Tim's income is 40 percent of Juan's income. To find out what percentage less Tim's income is than Juan's income, we subtract Tim's percentage from 100%: Percentage less = 100% - 40% Percentage less = 60% So, Tim's income is boxed{60} percent less than Juan's income.

answer:To solve for overrightarrow{a} and overrightarrow{b}, we start with the given equations: 1. overrightarrow{a}+overrightarrow{b}=(2,-4) 2. 3overrightarrow{a}-overrightarrow{b}=(-10,16) From equation 1, we can express overrightarrow{b} in terms of overrightarrow{a}: overrightarrow{b} = (2,-4) - overrightarrow{a} Substituting this expression for overrightarrow{b} into equation 2 gives us: 3overrightarrow{a} - ((2,-4) - overrightarrow{a}) = (-10,16) Simplifying this equation: 4overrightarrow{a} - (2,-4) = (-10,16) 4overrightarrow{a} = (-12,20) overrightarrow{a} = frac{(-12,20)}{4} overrightarrow{a} = (-3,5) However, there seems to be a mistake in the original solution provided. Let's correct this and find the accurate values for overrightarrow{a} and overrightarrow{b} by solving the system of equations correctly: Given: 1. overrightarrow{a}+overrightarrow{b}=(2,-4) 2. 3overrightarrow{a}-overrightarrow{b}=(-10,16) Adding these two equations to eliminate overrightarrow{b}: (overrightarrow{a}+overrightarrow{b}) + (3overrightarrow{a}-overrightarrow{b}) = (2,-4) + (-10,16) 4overrightarrow{a} = (-8,12) overrightarrow{a} = frac{(-8,12)}{4} overrightarrow{a} = (-2,3) Now, substituting overrightarrow{a} = (-2,3) into the first equation to find overrightarrow{b}: overrightarrow{a}+overrightarrow{b}=(2,-4) (-2,3) + overrightarrow{b} = (2,-4) overrightarrow{b} = (2,-4) - (-2,3) overrightarrow{b} = (4,-7) Now, to find overrightarrow{a}cdot overrightarrow{b}: overrightarrow{a}cdot overrightarrow{b} = (-2,3) cdot (4,-7) overrightarrow{a}cdot overrightarrow{b} = (-2)cdot(4) + (3)cdot(-7) overrightarrow{a}cdot overrightarrow{b} = -8 - 21 overrightarrow{a}cdot overrightarrow{b} = -29 Therefore, the correct answer is boxed{C}.