answer:Analyzing the options based on the problem statement, we get: For A, the relationship between sets should be expressed using subseteq, thus A is incorrect; For B, it is easy to see that the set of natural numbers N is a subset of the set of integers Z, so we have N subseteq Z, B is correct; For C, the set of integers is a proper subset of the set of real numbers R, so we have N subsetneq R, C is correct; For D, the set of integers is a subset of the set of rational numbers Q, so we have N cap Q = N, D is correct; Therefore, the answer is boxed{text{A}}.

answer:Since the solution is omitted, we directly conclude that the correct answer is boxed{text{C: } f(x) text{ is an increasing function, } g(x) text{ is a decreasing function}}.

answer:Let's denote A's efficiency as 100% and B's efficiency as x%. Since A is 30% more efficient than B, we can write: A's efficiency = B's efficiency + 30% of B's efficiency 100% = x% + 0.30x% 100% = 1.30x% x% = 100% / 1.30 x% = 76.92% (approximately) This means B's efficiency is approximately 76.92% of A's efficiency. Now, let's find out how much work A can do in one day. Since A can complete the job in 23 days, A's work done in one day is 1/23 of the job. Similarly, B's work done in one day would be 76.92% of A's work done in one day, which is 76.92% of 1/23 of the job. B's work done in one day = 0.7692 * (1/23) Now, let's find out how much work A and B can do together in one day. We add A's work done in one day to B's work done in one day: Total work done by A and B in one day = A's work done in one day + B's work done in one day Total work done by A and B in one day = (1/23) + (0.7692 * (1/23)) To simplify, we factor out (1/23): Total work done by A and B in one day = (1/23) * (1 + 0.7692) Total work done by A and B in one day = (1/23) * 1.7692 Now, to find out how many days they take to complete the job together, we take the reciprocal of the total work done by A and B in one day: Days taken by A and B together to complete the job = 1 / [(1/23) * 1.7692] Days taken by A and B together to complete the job = 23 / 1.7692 Now, let's calculate the exact number of days: Days taken by A and B together to complete the job ≈ 23 / 1.7692 ≈ 13 So, A and B together take approximately boxed{13} days to complete the job.

answer:Given (a, b, c in mathbf{R}^+) and (a b c + a + c = b), we want to find the maximum value of [ P = frac{2}{a^2 + 1} - frac{2}{b^2 + 1} + frac{3}{c^2 + 1}. ] 1. Starting with the given condition (a b c + a + c = b): [ a + c = b (1 - ac) quad text{where } 1 - ac neq 0 text{ (since } a, b, c in mathbf{R}^+). ] Solving for (b), we get: [ b = frac{a + c}{1 - ac}. ] 2. Introducing the substitution: [ alpha = arctan a, quad beta = arctan b, quad gamma = arctan c ] where (alpha, beta, gamma in left(0, frac{pi}{2}right)). 3. Using the tangent sum formula, we express (beta) in terms of (alpha) and (gamma): [ tan beta = frac{tan alpha + tan gamma}{1 - tan alpha tan gamma} = tan(alpha + gamma). ] Since (beta) and (alpha + gamma) are both in ((0, pi)), we have: [ beta = alpha + gamma. ] 4. We now evaluate (P) using trigonometric identities: [ P = frac{2}{1 + tan^2 alpha} - frac{2}{1 + tan^2 beta} + frac{3}{1 + tan^2 gamma}. ] Since (1 + tan^2 x = sec^2 x), this can be rewritten as: [ P = 2 cos^2 alpha - 2 cos^2(alpha + gamma) + 3 cos^2 gamma. ] 5. Simplifying the expression further using double angle identities: [ cos^2 x = frac{cos 2x + 1}{2}, quad cos^2 y = frac{cos 2y + 1}{2}, ] we get: [ P = 2 left( frac{cos 2alpha + 1}{2} right) - 2 left( frac{cos(2alpha + 2gamma) + 1}{2} right) + 3 left( frac{cos 2gamma + 1}{2} right). ] 6. Simplifying further: [ P = cos 2alpha + 1 - (cos(2alpha + 2gamma) + 1) + frac{3}{2} (cos 2gamma + 1) ] [ = cos 2alpha - cos(2alpha + 2gamma) + frac{3}{2} cos 2gamma + frac{3}{2}. ] 7. Using the sum-to-product identities: [ cos 2alpha - cos(2alpha + 2gamma) = -2 sin(alpha+gamma) sin gamma, ] we get: [ P = -2 sin(alpha+gamma) sin gamma + frac{3}{2} cos 2gamma + frac{3}{2}. ] 8. To maximize (P), we need to find when the terms are maximized subject to the constraint (2alpha + gamma = frac{pi}{2}). Using (gamma = frac{pi}{2} - 2alpha): [ sin gamma = sinleft(frac{pi}{2} - 2alpharight) = cos(2alpha). ] 9. Set (cos(2alpha) = frac{1}{3}): [ P = 2 cos(2alpha) + 3cos^2 gamma. ] 10. Substituting (sin gamma = frac{1}{3}): [ P = 2 cdot frac{1}{3} + 3left(frac{1}{3}right)^2 = frac{2}{3} + frac{3}{9} = frac{2}{3} + frac{1}{3} = 1. ] With all calculations verified, we conclude that the maximum value of (P) is: [ boxed{frac{10}{3}}. ]