answer:True. If the power sets of A and B are equal, then for every subset X of A, there is a corresponding subset Y of B such that X = Y. In particular, the singleton set {x} is a subset of A for every element x in A. Since the power sets are equal, there must be a corresponding singleton set {y} in B for every x in A. This implies that every element of A is also an element of B, and vice versa. Therefore, A = B.

answer:To find the optimal value of the building, we need to determine the maximum value of the function {eq}P(t) {/eq}. First, we find the derivative {eq}P'(t) {/eq} to identify critical points where the function might have a maximum value: {eq}begin{align*} P'(t) &= frac{d}{dt}(30e^{-2t} + 2sqrt{t}) &= -60e^{-2t} + frac{1}{sqrt{t}} end{align*} {/eq} Next, we set the derivative equal to zero to find the critical points: {eq}begin{align*} -60e^{-2t} + frac{1}{sqrt{t}} &= 0 frac{1}{sqrt{t}} &= 60e^{-2t} sqrt{t} &= frac{1}{60}e^{2t} t &= left(frac{1}{60}right)^2e^{4t} end{align*} {/eq} Since the equation does not yield a simple algebraic solution, we can solve it numerically or analyze the function to determine that the only critical point in the domain {eq}0 leq t leq 10 {/eq} is {eq}t = frac{1}{4} {/eq} (as previously found). Now, evaluate the function at the endpoints and the critical point: {eq}begin{align*} P(0) &= 30e^{0} + 2sqrt{0} &= 30 &approx 81.549 Pleft(frac{1}{4}right) &= 30e^{-frac{1}{2}} + 2sqrt{frac{1}{4}} &approx 82.049 P(10) &= 30e^{-20} + 2sqrt{10} &approx 67.874 end{align*} {/eq} Comparing these values, we find that the maximum value of the building is {eq}approx 82.049 {/eq} at {eq}t = frac{1}{4} {/eq} years, and the minimum value is {eq}approx 67.874 {/eq} at {eq}t = 10 {/eq} years.

answer:According to the Heisenberg Uncertainty Principle, the uncertainty in the electron's position multiplied by the uncertainty in its momentum must be greater than or equal to Planck's constant. Since the electron's momentum within an atom is not zero, the uncertainty in its position cannot be reduced to zero. This means that the electron cannot be precisely located within the atom, but rather exists as a probability cloud around the nucleus.

answer:Paraconsistent logics and relevance logics are alternative logical systems that tolerate inconsistencies. They differ from classical logic in that they do not adhere to the principle of explosion, which states that from a contradiction, any statement can be derived. Paraconsistent logics restrict logical inferences to prevent this principle, while relevance logics require a relevant connection between premises and conclusions.