answer：First, determine how much of the pizza Peter ate on his own. He ate 2 slices out of 16, which is ( frac{2}{16} ).Next, calculate the amount Peter shared with his brother. They shared 2 slices equally, so each ate 1 slice. This is ( frac{1}{16} ) of the pizza.Finally, add the two amounts together to find the total fraction of the pizza Peter ate:[frac{2}{16} + frac{1}{16} = frac{3}{16}]

answer：Okay, so I need to find the remainder when the polynomial ( x^6 - 2x^5 + x^4 - x^2 + 3x - 1 ) is divided by ( (x^2 - 1)(x + 2) ). Hmm, let me think about how to approach this.First, I remember that when dividing a polynomial by another polynomial, the remainder has a degree less than the divisor. The divisor here is ( (x^2 - 1)(x + 2) ). Let me check the degree of this divisor. ( x^2 - 1 ) is a quadratic, so degree 2, and ( x + 2 ) is linear, degree 1. Multiplying them together, the degree is 2 + 1 = 3. So, the divisor is a cubic polynomial, which means the remainder should be a quadratic polynomial, right? So, the remainder will be of the form ( ax^2 + bx + c ).Alright, so I can write the division as:[ x^6 - 2x^5 + x^4 - x^2 + 3x - 1 = (x^2 - 1)(x + 2)q(x) + ax^2 + bx + c ]where ( q(x) ) is the quotient polynomial.Now, to find ( a ), ( b ), and ( c ), I can use the fact that the divisor ( (x^2 - 1)(x + 2) ) has roots at ( x = 1 ), ( x = -1 ), and ( x = -2 ). That means when I substitute these values into the equation, the left side will equal the right side, which simplifies because the divisor becomes zero.Let me start with ( x = 1 ):Plugging into the original polynomial:[ 1^6 - 2(1)^5 + 1^4 - 1^2 + 3(1) - 1 = 1 - 2 + 1 - 1 + 3 - 1 = 1 ]On the right side, since ( (1^2 - 1)(1 + 2) = 0 ), it becomes ( a(1)^2 + b(1) + c = a + b + c ).So, equation 1: ( a + b + c = 1 ).Next, ( x = -1 ):Plugging into the original polynomial:[ (-1)^6 - 2(-1)^5 + (-1)^4 - (-1)^2 + 3(-1) - 1 = 1 + 2 + 1 - 1 - 3 - 1 = -1 ]On the right side, ( (-1)^2 - 1 = 0 ), so it becomes ( a(-1)^2 + b(-1) + c = a - b + c ).So, equation 2: ( a - b + c = -1 ).Now, ( x = -2 ):Plugging into the original polynomial:[ (-2)^6 - 2(-2)^5 + (-2)^4 - (-2)^2 + 3(-2) - 1 ]Let me compute each term:- ( (-2)^6 = 64 )- ( -2(-2)^5 = -2(-32) = 64 )- ( (-2)^4 = 16 )- ( -(-2)^2 = -4 )- ( 3(-2) = -6 )- ( -1 )Adding them up: 64 + 64 + 16 - 4 - 6 - 1 = 64 + 64 is 128, plus 16 is 144, minus 4 is 140, minus 6 is 134, minus 1 is 133. Wait, that seems high. Let me double-check:- ( (-2)^6 = 64 )- ( -2(-2)^5 = -2*(-32) = 64 )- ( (-2)^4 = 16 )- ( -(-2)^2 = -4 )- ( 3*(-2) = -6 )- ( -1 )So, 64 + 64 = 128, 128 + 16 = 144, 144 - 4 = 140, 140 - 6 = 134, 134 -1 = 133. Hmm, that's correct.On the right side, substituting ( x = -2 ) into the remainder: ( a(-2)^2 + b(-2) + c = 4a - 2b + c ).So, equation 3: ( 4a - 2b + c = 133 ). Wait, that can't be right because when I plug ( x = -2 ) into the original polynomial, I get 133, but the divisor at ( x = -2 ) is ( ((-2)^2 -1)(-2 + 2) = (4 -1)(0) = 0 ), so the right side is just the remainder, which is 133. But that seems way too big. Did I make a mistake in calculation?Wait, let me recalculate the original polynomial at ( x = -2 ):[ (-2)^6 = 64 ][ -2(-2)^5 = -2*(-32) = 64 ][ (-2)^4 = 16 ][ -(-2)^2 = -4 ][ 3*(-2) = -6 ][ -1 ]So, adding up: 64 + 64 = 128, 128 + 16 = 144, 144 - 4 = 140, 140 -6 = 134, 134 -1 = 133. Hmm, that's correct. So equation 3 is ( 4a - 2b + c = 133 ).Wait, but earlier when I thought about the remainder, I thought it would be a quadratic, but 133 is a constant term. That seems inconsistent because the remainder is supposed to be quadratic. Maybe I made a mistake in the substitution?Wait, no. The remainder is quadratic, so when I plug in ( x = -2 ), it's just evaluating the quadratic at that point, so it can result in a large number. So, 133 is correct.So, now I have three equations:1. ( a + b + c = 1 )2. ( a - b + c = -1 )3. ( 4a - 2b + c = 133 )Let me try to solve these equations step by step.First, subtract equation 2 from equation 1:( (a + b + c) - (a - b + c) = 1 - (-1) )Simplify:( a + b + c - a + b - c = 2 )Which simplifies to:( 2b = 2 )So, ( b = 1 ).Now, substitute ( b = 1 ) into equation 1:( a + 1 + c = 1 )Which simplifies to:( a + c = 0 )So, ( c = -a ).Now, substitute ( b = 1 ) and ( c = -a ) into equation 3:( 4a - 2(1) + (-a) = 133 )Simplify:( 4a - 2 - a = 133 )Which is:( 3a - 2 = 133 )Add 2 to both sides:( 3a = 135 )Divide by 3:( a = 45 )Then, since ( c = -a ), ( c = -45 ).So, the remainder is ( 45x^2 + x - 45 ).Wait, but when I plug ( x = 1 ) into this remainder, I get ( 45 + 1 - 45 = 1 ), which matches equation 1. For ( x = -1 ), it's ( 45 - 1 - 45 = -1 ), which matches equation 2. And for ( x = -2 ), it's ( 45*4 + (-2) -45 = 180 -2 -45 = 133 ), which matches equation 3. So, it seems correct.But wait, earlier I thought the remainder would be ( frac{7}{3}x^2 + x - frac{7}{3} ), but that was in a different problem. In this case, the calculations led me to ( 45x^2 + x - 45 ). That seems correct based on the equations I solved.Wait, but let me double-check the original polynomial at ( x = -2 ). Maybe I made a mistake there.Original polynomial: ( x^6 - 2x^5 + x^4 - x^2 + 3x - 1 ).At ( x = -2 ):( (-2)^6 = 64 )( -2*(-2)^5 = -2*(-32) = 64 )( (-2)^4 = 16 )( -(-2)^2 = -4 )( 3*(-2) = -6 )( -1 )Adding up: 64 + 64 = 128, 128 + 16 = 144, 144 - 4 = 140, 140 -6 = 134, 134 -1 = 133. Yes, that's correct.So, the remainder is ( 45x^2 + x - 45 ).Wait, but that seems like a very large coefficient. Maybe I made a mistake in the setup.Wait, let me think again. The divisor is ( (x^2 - 1)(x + 2) ), which is ( x^3 + 2x^2 - x - 2 ). So, when I perform polynomial division, the remainder should be of degree less than 3, which is correct.But perhaps I should try another method, like polynomial long division, to verify.Alternatively, maybe I can use the Remainder Theorem in a different way.Wait, another approach is to express the original polynomial as ( (x^2 -1)(x + 2)q(x) + ax^2 + bx + c ), and then equate coefficients.But that might be more involved. Alternatively, I can use the fact that the remainder when divided by ( (x - 1) ), ( (x + 1) ), and ( (x + 2) ) must satisfy certain conditions.Wait, but I already used the roots to set up the equations, so that should be correct.Wait, but in the initial problem statement, the user wrote the same problem and got a different answer, ( frac{7}{3}x^2 + x - frac{7}{3} ). But in my calculation, I got 45x² + x -45. That's a big difference. So, perhaps I made a mistake in the calculation.Wait, let me check the original polynomial again. The user wrote: ( x^6 - 2x^5 + x^4 - x^2 + 3x - 1 ). Yes, that's correct.Wait, when I plugged in ( x = -2 ), I got 133. But if I use the remainder ( 45x^2 + x -45 ), then at ( x = -2 ), it's 45*(4) + (-2) -45 = 180 -2 -45 = 133, which matches. So, that seems correct.But in the initial problem, the user had a different answer. Maybe they had a different polynomial or divisor.Wait, looking back, the user wrote: "Find the remainder when ( x^6 - 2x^5 + x^4 - x^2 + 3x - 1 ) is divided by ( (x^2 - 1)(x + 2) )." So, same as mine.But in their solution, they got ( frac{7}{3}x^2 + x - frac{7}{3} ). But in my calculation, I got 45x² + x -45. That's a big discrepancy. So, perhaps I made a mistake in the calculation.Wait, let me check the equations again.Equation 1: ( a + b + c = 1 )Equation 2: ( a - b + c = -1 )Equation 3: ( 4a - 2b + c = 133 )Subtracting equation 2 from equation 1: ( 2b = 2 ) so ( b = 1 ). Correct.Then, equation 1: ( a + 1 + c = 1 ) so ( a + c = 0 ). So, ( c = -a ). Correct.Then, equation 3: ( 4a - 2*1 + (-a) = 133 ) → ( 4a -2 -a = 133 ) → ( 3a -2 = 133 ) → ( 3a = 135 ) → ( a = 45 ). So, ( c = -45 ). So, remainder is ( 45x² + x -45 ).Wait, but that seems correct. So, why does the initial problem have a different answer? Maybe I misread the problem.Wait, in the initial problem, the user wrote the same problem, but in their solution, they got ( frac{7}{3}x^2 + x - frac{7}{3} ). But in my calculation, I got 45x² + x -45. That's a big difference. So, perhaps I made a mistake in the calculation.Wait, let me check the original polynomial at ( x = -2 ) again.Original polynomial: ( x^6 - 2x^5 + x^4 - x^2 + 3x - 1 ).At ( x = -2 ):( (-2)^6 = 64 )( -2*(-2)^5 = -2*(-32) = 64 )( (-2)^4 = 16 )( -(-2)^2 = -4 )( 3*(-2) = -6 )( -1 )Adding up: 64 + 64 = 128, 128 + 16 = 144, 144 -4 = 140, 140 -6 = 134, 134 -1 = 133. Yes, that's correct.So, equation 3 is ( 4a -2b + c = 133 ). Correct.So, solving the equations gives ( a = 45 ), ( b = 1 ), ( c = -45 ). So, the remainder is ( 45x² + x -45 ).But in the initial problem, the user got ( frac{7}{3}x^2 + x - frac{7}{3} ). So, perhaps they made a mistake, or I misread the problem.Wait, maybe the original polynomial was different. Let me check.The user wrote: "Find the remainder when ( x^6 - 2x^5 + x^4 - x^2 + 3x - 1 ) is divided by ( (x^2 - 1)(x + 2) )."Yes, same as mine.Wait, perhaps I made a mistake in the setup. Let me try another approach.Let me perform polynomial long division to verify.Divide ( x^6 - 2x^5 + x^4 - x^2 + 3x - 1 ) by ( (x^3 + 2x^2 - x - 2) ).Wait, because ( (x^2 -1)(x + 2) = x^3 + 2x^2 -x -2 ).So, let's set up the division:Dividend: ( x^6 - 2x^5 + x^4 + 0x^3 - x^2 + 3x -1 )Divisor: ( x^3 + 2x^2 -x -2 )We need to find the quotient and remainder.First term: ( x^6 / x^3 = x^3 ). Multiply divisor by ( x^3 ):( x^6 + 2x^5 -x^4 -2x^3 )Subtract from dividend:( (x^6 - 2x^5 + x^4 + 0x^3 - x^2 + 3x -1 ) - (x^6 + 2x^5 -x^4 -2x^3) )= ( -4x^5 + 2x^4 + 2x^3 -x^2 + 3x -1 )Next term: ( -4x^5 / x^3 = -4x^2 ). Multiply divisor by ( -4x^2 ):( -4x^5 -8x^4 +4x^3 +8x^2 )Subtract:( (-4x^5 + 2x^4 + 2x^3 -x^2 + 3x -1 ) - (-4x^5 -8x^4 +4x^3 +8x^2) )= ( 10x^4 -2x^3 -9x^2 + 3x -1 )Next term: ( 10x^4 / x^3 = 10x ). Multiply divisor by ( 10x ):( 10x^4 +20x^3 -10x^2 -20x )Subtract:( (10x^4 -2x^3 -9x^2 + 3x -1 ) - (10x^4 +20x^3 -10x^2 -20x) )= ( -22x^3 + x^2 +23x -1 )Next term: ( -22x^3 / x^3 = -22 ). Multiply divisor by ( -22 ):( -22x^3 -44x^2 +22x +44 )Subtract:( (-22x^3 + x^2 +23x -1 ) - (-22x^3 -44x^2 +22x +44) )= ( 45x^2 + x -45 )So, the remainder is ( 45x^2 + x -45 ), which matches my earlier result.Therefore, the remainder is ( 45x^2 + x -45 ).Wait, but in the initial problem, the user got ( frac{7}{3}x^2 + x - frac{7}{3} ). So, perhaps they made a mistake in their calculation.Alternatively, maybe I misread the original polynomial. Let me check again.Original polynomial: ( x^6 - 2x^5 + x^4 - x^2 + 3x - 1 ). Yes, that's correct.So, I think my calculation is correct, and the remainder is ( 45x^2 + x -45 ).But wait, in the initial problem, the user wrote the same problem and got a different answer. Maybe they made a mistake in their calculation.Alternatively, perhaps I made a mistake in the long division. Let me check the long division again.Dividend: ( x^6 - 2x^5 + x^4 + 0x^3 - x^2 + 3x -1 )Divisor: ( x^3 + 2x^2 -x -2 )First term: ( x^6 / x^3 = x^3 ). Multiply divisor by ( x^3 ):( x^6 + 2x^5 -x^4 -2x^3 )Subtract:( (x^6 - 2x^5 + x^4 + 0x^3 - x^2 + 3x -1 ) - (x^6 + 2x^5 -x^4 -2x^3) )= ( -4x^5 + 2x^4 + 2x^3 -x^2 + 3x -1 )Next term: ( -4x^5 / x^3 = -4x^2 ). Multiply divisor by ( -4x^2 ):( -4x^5 -8x^4 +4x^3 +8x^2 )Subtract:( (-4x^5 + 2x^4 + 2x^3 -x^2 + 3x -1 ) - (-4x^5 -8x^4 +4x^3 +8x^2) )= ( 10x^4 -2x^3 -9x^2 + 3x -1 )Next term: ( 10x^4 / x^3 = 10x ). Multiply divisor by ( 10x ):( 10x^4 +20x^3 -10x^2 -20x )Subtract:( (10x^4 -2x^3 -9x^2 + 3x -1 ) - (10x^4 +20x^3 -10x^2 -20x) )= ( -22x^3 + x^2 +23x -1 )Next term: ( -22x^3 / x^3 = -22 ). Multiply divisor by ( -22 ):( -22x^3 -44x^2 +22x +44 )Subtract:( (-22x^3 + x^2 +23x -1 ) - (-22x^3 -44x^2 +22x +44) )= ( 45x^2 + x -45 )Yes, that's correct. So, the remainder is indeed ( 45x^2 + x -45 ).Therefore, I think the initial problem's solution was incorrect, and the correct remainder is ( 45x^2 + x -45 ).

answer：Alright, so I've got this problem here about a substance that absorbs moisture, increasing its mass. The problem states that to absorb 1400 kg of moisture, you need to take an undivided substance which is 300 kg more than the divided substance. Then, it asks for the percentage of the substance's mass that the absorbed moisture represents in both the divided and undivided cases. It also mentions that in the second case (undivided), this percentage is 105 less than in the first case (divided).Okay, let's break this down. First, I need to figure out what exactly is being asked. We have two scenarios: one where the substance is divided and another where it's undivided. In both cases, the substance absorbs 1400 kg of moisture. The undivided substance is 300 kg more than the divided one. So, if I let the mass of the divided substance be x kg, then the undivided substance would be x + 300 kg.Now, the problem is about percentages. Specifically, it's asking for the percentage of the substance's mass that the absorbed moisture represents in both cases. Let's denote the percentage for the divided substance as P1 and for the undivided substance as P2. It also says that P2 is 105 less than P1. So, P1 = P2 + 105.But wait, percentages can't be more than 100% in some contexts, but in this case, since the substance is absorbing moisture and increasing its mass, the percentage could be more than 100%. So, it's possible for P1 and P2 to be greater than 100%.Alright, so let's formalize this. For the divided substance:Mass of substance = x kgMass of absorbed moisture = 1400 kgPercentage P1 = (1400 / x) * 100For the undivided substance:Mass of substance = x + 300 kgMass of absorbed moisture = 1400 kgPercentage P2 = (1400 / (x + 300)) * 100And we know that P1 = P2 + 105So, substituting the expressions for P1 and P2:(1400 / x) * 100 = (1400 / (x + 300)) * 100 + 105Hmm, okay. Let's simplify this equation. First, let's divide both sides by 100 to make it simpler:1400 / x = 1400 / (x + 300) + 1.05Wait, 105 divided by 100 is 1.05, right? So, that's correct.Now, let's write it as:1400 / x - 1400 / (x + 300) = 1.05To solve for x, we can find a common denominator for the left side:[1400(x + 300) - 1400x] / [x(x + 300)] = 1.05Simplifying the numerator:1400x + 420000 - 1400x = 420000So, the equation becomes:420000 / [x(x + 300)] = 1.05Now, multiply both sides by x(x + 300):420000 = 1.05 * x(x + 300)Let's divide both sides by 1.05 to simplify:420000 / 1.05 = x(x + 300)Calculating 420000 / 1.05:420000 / 1.05 = 400000So, we have:400000 = x(x + 300)Expanding the right side:x^2 + 300x = 400000Bringing all terms to one side:x^2 + 300x - 400000 = 0Now, we have a quadratic equation. Let's solve for x using the quadratic formula:x = [-b ± sqrt(b^2 - 4ac)] / (2a)Here, a = 1, b = 300, c = -400000Calculating the discriminant:b^2 - 4ac = 300^2 - 4*1*(-400000) = 90000 + 1600000 = 1690000Square root of 1690000 is 1300.So, x = [-300 ± 1300] / 2We have two solutions:x = (-300 + 1300)/2 = 1000/2 = 500x = (-300 - 1300)/2 = -1600/2 = -800Since mass can't be negative, we discard x = -800.So, x = 500 kgTherefore, the divided substance is 500 kg, and the undivided substance is 500 + 300 = 800 kg.Now, let's calculate the percentages.For the divided substance:P1 = (1400 / 500) * 100 = 280%For the undivided substance:P2 = (1400 / 800) * 100 = 175%And indeed, 280% - 175% = 105%, which matches the condition given in the problem.So, the percentages are 280% for the divided substance and 175% for the undivided substance.

answer：First, I need to understand the relationship between the efficiencies of A and B. Since A is 30% more efficient than B, I can express A's efficiency in terms of B's efficiency.Let’s denote B's efficiency as ( B ). Therefore, A's efficiency would be ( B + 0.30B = 1.30B ).Next, I'll assume the total work to be completed is ( W ). When A and B work together, their combined efficiency is ( A + B = 1.30B + B = 2.30B ).They complete the work in 13 days, so the total work can be expressed as:[W = (A + B) times 13 = 2.30B times 13 = 29.90B]Now, to find out how many days A alone would take to complete the same work, I'll use A's efficiency:[text{Days for A alone} = frac{W}{A} = frac{29.90B}{1.30B} = 23 text{ days}]Therefore, A alone would take 23 days to complete the job.

answer：Okay, so I have this problem here about a geometric sequence. The sum of the first (n) terms is given by (S_n = 3^n + r), and I need to find the value of (r). The options are A: 0, B: -1, C: 1, D: 3. Hmm, let me think about how to approach this.First, I remember that in a geometric sequence, the sum of the first (n) terms can be expressed using the formula (S_n = a_1 frac{r^n - 1}{r - 1}), where (a_1) is the first term and (r) is the common ratio. But wait, in this problem, they've given the sum as (S_n = 3^n + r). That seems a bit different from the standard formula. Maybe I can use the given expression to find the common ratio and the first term.Let me recall that the (n)-th term of a geometric sequence can be found by subtracting the sum of the first (n-1) terms from the sum of the first (n) terms. So, (a_n = S_n - S_{n-1}). Let me try that.Given (S_n = 3^n + r), then (S_{n-1} = 3^{n-1} + r). So, subtracting these, we get:[a_n = S_n - S_{n-1} = (3^n + r) - (3^{n-1} + r) = 3^n - 3^{n-1}]Simplify that:[3^n - 3^{n-1} = 3^{n-1}(3 - 1) = 2 cdot 3^{n-1}]So, the (n)-th term is (2 cdot 3^{n-1}). That looks like a geometric sequence with the first term (a_1 = 2) and common ratio (3), right? Because each term is 3 times the previous term.Wait, let me check that. If (a_1 = 2), then (a_2 = 2 cdot 3 = 6), (a_3 = 6 cdot 3 = 18), and so on. So, the sequence is 2, 6, 18, 54, etc. Let me see if this matches the sum given.The sum of the first term (S_1) should be (a_1 = 2). According to the given formula, (S_1 = 3^1 + r = 3 + r). So, setting that equal to 2:[3 + r = 2 implies r = 2 - 3 = -1]So, (r = -1). That would be option B. Hmm, let me verify this with another term to make sure.Let's compute (S_2). According to the given formula, (S_2 = 3^2 + (-1) = 9 - 1 = 8). On the other hand, the sum of the first two terms of the sequence is (a_1 + a_2 = 2 + 6 = 8). That matches.Let's check (S_3). The formula gives (3^3 + (-1) = 27 - 1 = 26). The sum of the first three terms is (2 + 6 + 18 = 26). Perfect, that's consistent too.Just to be thorough, let me check (S_4). The formula gives (3^4 + (-1) = 81 - 1 = 80). The sum of the first four terms is (2 + 6 + 18 + 54 = 80). Yep, that works as well.So, it seems like (r = -1) is indeed the correct answer. I think I've got it.**Final Answer**boxed{B}

answer：Okay, so I have this logic problem here about parachuting training. Let me try to understand it step by step. First, there are two people, A and B, who each make a jump. The problem defines two propositions: P represents "Person A lands in the designated area," and q represents "Person B lands in the designated area." The statement we need to translate into logical notation is "At least one person lands in the designated area." The options given are:A: ¬p ∨ ¬q B: p ∨ ¬q C: ¬p ∧ ¬q D: p ∨ q Alright, so I need to figure out which of these represents "At least one person lands in the designated area." Let me break it down.First, let's recall what "at least one" means in logic. "At least one" is equivalent to saying that either one of them happens, or both. In logical terms, this is an inclusive OR. So, if I have two propositions, P and q, then "at least one of P or q is true" would be P ∨ q. Let me verify that. If P is true, meaning person A lands in the designated area, then regardless of person B, the statement "at least one" is satisfied. Similarly, if q is true, meaning person B lands in the designated area, regardless of person A, the statement is satisfied. If both P and q are true, then of course, it's still satisfied. So, P ∨ q seems to capture the essence of "at least one."Now, let's look at the options again. Option D is p ∨ q, which matches what I just thought. But let me make sure I'm not missing anything by checking the other options.Option A is ¬p ∨ ¬q. This would mean "either person A does not land in the designated area, or person B does not land in the designated area." That's actually the opposite of what we want because it's talking about at least one not landing, not at least one landing. So, that's not correct.Option B is p ∨ ¬q. This translates to "either person A lands, or person B does not land." This isn't exactly "at least one lands" because it's combining a positive statement about A with a negative statement about B. It doesn't cover all cases where at least one lands. For example, if both A and B land, this statement would still be true because p is true, but it also includes cases where B doesn't land, which isn't necessary for "at least one." So, this seems a bit off.Option C is ¬p ∧ ¬q. This means "both person A does not land and person B does not land." That's the opposite of what we want. It's saying neither lands, which is the negation of "at least one lands." So, definitely not this one.So, going back, Option D is p ∨ q, which correctly represents "at least one person lands in the designated area." It covers all scenarios where either A lands, B lands, or both land. I think I've got it. Option D is the right answer. But just to be thorough, let me think about truth tables to confirm.If I construct a truth table for p ∨ q, it should be true in all cases except when both p and q are false. That is exactly the scenario where neither A nor B lands in the designated area, which is the only case where "at least one lands" would be false. So, yes, p ∨ q correctly represents the statement.Another way to look at it is through the lens of natural language. When someone says "at least one," they're saying that one or more of the options is true. In logic, that's an inclusive OR, which is exactly what p ∨ q represents.I don't think I've missed anything here. All other options either invert the meaning or only partially capture the intended statement. So, I'm confident that D is the correct choice.