Q: Syllogistic inference about carpeting and color: All offices on the 9th floor have wall-to-wall carpeting, and no wall-to-wall carpeting is pink; decide whether the conclusion "None of the 9th-floor offices has pink wall-to-wall carpeting" must foll…

Given data Premise 1: All 9th-floor offices have wall-to-wall carpeting. Premise 2: No wall-to-wall carpeting is pink. Conclusion to test: No 9th-floor office has pink wall-to-wall carpeting. Concept / Approach Use categorical logic: If all items in Group A have Property C, and nothing with Property C can be pink, then nothing in Group A has pink carpeting. Step-by-step reasoning Step 1: Every 9th-floor office ∈ {carpeted}.Step 2: {carpeted} ∩ {pink} = ∅ (by Premise 2, carpets cannot be pink).Step 3: Therefore, for any 9th-floor office, "pink wall-to-wall carpeting" is impossible. Verification / Alternative view Contrapositive thinking: If an office had pink wall-to-wall carpeting, then there would exist pink wall-to-wall carpeting, contradicting Premise 2. Common pitfalls Assuming some 9th-floor offices might lack carpets; Premise 1 rules that out. Final Answer Definitely true.

Question 1

Logic consistency check on comparative ages: given 'Tanya is older than Eric' and 'Cliff is older than Tanya', evaluate the truth of 'Eric is older than Cliff'

Accepted Answer

Given data T > E (Tanya older than Eric) C > T (Cliff older than Tanya) Concept/Approach (why this method) Use transitivity of the 'older than' relation: if C > T and T > E then C > E. Step-by-Step deduction 1) From C > T and T > E, conclude C > E.2) Statement to test says E > C, which directly contradicts C > E. Verification/Alternative No equalities allowed by 'older than'; contradiction is decisive. Common pitfalls Reversing the order or assuming non-transitive relations. Final Answer False

Question 2

Transitive price comparison reasoning: given 'Blueberries cost more than strawberries' and 'Blueberries cost less than raspberries', determine whether 'Raspberries cost more than strawberries and blueberries' is true

Accepted Answer

Given data B > S (Blueberries > Strawberries) R > B (since Blueberries < Raspberries) Concept/Approach (why this method) Apply order transitivity on 'cost more than': if R > B and B > S, then R > S and R > B. Step-by-Step deduction 1) From B > S and R > B, infer R > S.2) We already have R > B; combine to yield 'R > S and R > B'. Verification/Alternative Construct sample numbers (S=1, B=2, R=3) → statement holds. Common pitfalls Assuming 'and' requires extra information—here transitivity suffices. Final Answer True

Question 3

Syllogism validation with set inclusion: if all trees in the park are flowering and some trees are dogwoods, assess the claim 'All dogwoods in the park are flowering trees'

Accepted Answer

Given data All Trees_in_park ⊆ Flowering. Some Trees_in_park ∩ Dogwoods ≠ ∅. Concept/Approach (why this method) Universal inclusion: every member of the 'trees in the park' set is flowering; any subclass (e.g., dogwoods in the park) inherits this property. Step-by-Step deduction 1) Let D = {dogwoods in the park}. By premise, D ⊆ Trees_in_park.2) Since Trees_in_park ⊆ Flowering, transitivity gives D ⊆ Flowering. Verification/Alternative Venn diagram: place 'Trees' circle fully inside 'Flowering'; 'Dogwoods in park' is a sub-region within 'Trees', hence also inside 'Flowering'. Common pitfalls Confusing 'some trees are dogwoods' with 'some dogwoods are flowering'—here all park trees are flowering, so all park dogwoods are flowering. Final Answer True

Question 4

Speed ranking logic: given 'Mara runs faster than Gail' and 'Lily runs faster than Mara', evaluate the truth of 'Gail runs faster than Lily'

Accepted Answer

Given data M > G (Mara faster than Gail) L > M (Lily faster than Mara) Concept/Approach (why this method) Use strict order transitivity on 'faster than': L > M > G ⇒ L > G. Step-by-Step deduction 1) From L > M and M > G, deduce L > G.2) The statement claims G > L, which contradicts L > G. Verification/Alternative No equalities or ties are stated; 'faster than' is strict, so contradiction stands. Common pitfalls Overlooking transitivity; assuming cyclical speed relations (not possible under strict 'faster than'). Final Answer False

Question 5

Relative apartment pricing logic: with Riverdale < Gaslight Commons and Livingston Gate > Gaslight Commons, determine whether 'Livingston Gate costs the most among the three' is correct

Accepted Answer

Given data R < G (Riverdale cheaper than Gaslight Commons) L > G (Livingston Gate pricier than Gaslight Commons) Concept/Approach (why this method) Order chaining: if L > G and G > R, then L > G and L > R → L is the most expensive of the three. Step-by-Step deduction 1) From R < G, we have G > R.2) Combine with L > G ⇒ L > G and L > R.3) Therefore, Livingston Gate is the costliest among {R, G, L}. Verification/Alternative Plug sample numbers (R=10, G=15, L=20) → statement holds. Common pitfalls Thinking we need a direct L vs. R statement; it is implied via transitivity. Final Answer True

Question 6

Mall size comparison logic: given 'Kingston has more stores than the Galleria' and 'Four Corners has fewer stores than the Galleria', evaluate 'Kingston has more stores than Four Corners'

Accepted Answer

Given data K > G (Kingston > Galleria) F < G (Four Corners < Galleria) Concept/Approach (why this method) Transitivity on inequalities: if K > G and G > F, then K > F. Step-by-Step deduction 1) From F < G, infer G > F.2) With K > G, chain to get K > F. Verification/Alternative Example counts (K=120, G=100, F=80) satisfy all statements and confirm the conclusion. Common pitfalls Assuming we need exact store counts; ordinal relations suffice. Final Answer True

Question 7

Logical reasoning (premises-to-conclusion): All the tulips in Zoe's garden are white; all the pansies in Zoe's garden are yellow; determine whether the concluding statement—"All the flowers in Zoe's garden are either white or yellow"—must be true ba…

Accepted Answer

Given data Premise 1: All tulips in Zoe's garden are white. Premise 2: All pansies in Zoe's garden are yellow. Candidate conclusion: All flowers in Zoe's garden are either white or yellow. Concept / Approach We are testing whether the conclusion is logically entailed by the premises (i.e., must be true if the premises are true), not whether it could happen to be true in Zoe's actual garden.Premises mention only two categories (tulips, pansies) and their colors; they say nothing about any other flowers that might also be in the garden. Step-by-step reasoning Step 1: From Premise 1, every tulip is white. This restricts tulips to the set {white}.Step 2: From Premise 2, every pansy is yellow. This restricts pansies to the set {yellow}.Step 3: The premises do not claim that tulips and pansies are the only flowers in the garden; other kinds (e.g., roses, lilies) may exist.Step 4: If other flower types exist, they could be red, pink, purple, etc. Therefore, it does not logically follow that all flowers are either white or yellow.Step 5: Because the existence and colors of other flower types are unspecified, the conclusion is not guaranteed. Verification / Alternative view Construct a countermodel: Suppose Zoe also grows red roses. Premises 1 and 2 remain true, but then not all flowers are white or yellow. Hence the conclusion does not necessarily follow. Common pitfalls Assuming that only tulips and pansies exist because they were the only types mentioned. Confusing "could be true" with "must be true." Logical entailment requires necessity, not possibility. Final Answer Cannot be determined from the given premises—the conclusion is not logically entailed.

Question 8

Comparative reasoning about quantities watched: Josh saw more movies than Stephen, and Stephen saw fewer movies than Darren; decide, strictly from these two premises, whether the statement "Darren saw more movies than Josh" must be true.

Accepted Answer

Given data Premise 1: Josh > Stephen (Josh saw more movies than Stephen). Premise 2: Darren > Stephen (Stephen saw fewer than Darren). Claim to test: Darren > Josh. Concept / Approach Translate the sentences into inequalities and then compare what is known and unknown.Knowing that both Josh and Darren exceed Stephen does not, by itself, order Josh and Darren relative to each other. Step-by-step reasoning Step 1: From Premise 1, J > S.Step 2: From Premise 2, D > S.Step 3: There is no premise comparing J and D directly. Both could be greater than S, but either one could be larger.Step 4: Construct models: (a) J=12, S=10, D=15 ⇒ D > J (claim true). (b) J=16, S=10, D=15 ⇒ J > D (claim false). Since both are compatible with the premises, the claim is not entailed. Verification / Alternative view Because two different consistent assignments yield opposite truth values for the claim, the truth of the claim is undetermined from the premises. Common pitfalls Assuming transitivity via Stephen (e.g., "greater than Stephen" implies a chain between Josh and Darren). It does not. Confusing "possible" with "necessary." Final Answer Cannot be determined from the given premises.

Question 9

Weight-ordering logic: Rover weighs less than Fido, and Rover weighs more than Boomer; determine whether the conclusion "Of the three dogs, Boomer weighs the least" must logically follow from the two given premises.

Accepted Answer

Given data Premise 1: Rover < Fido. Premise 2: Rover > Boomer. Claim: Boomer is the lightest of the three (i.e., Boomer < Rover and Boomer < Fido). Concept / Approach Chain the inequalities to find a strict ordering. Step-by-step reasoning Step 1: From Premise 2, Boomer < Rover.Step 2: From Premise 1, Rover < Fido.Step 3: Combine the two: Boomer < Rover < Fido. This establishes a strict ascending order of weights.Step 4: Therefore Boomer is lighter than Rover and also lighter than Fido, so Boomer is the least. Verification / Alternative view Any numeric assignment respecting the premises (e.g., Boomer=10, Rover=12, Fido=14) makes the conclusion true. There is no counterexample consistent with the premises. Common pitfalls Overthinking ties: the strict inequalities ("less than", "more than") already exclude equality. Final Answer Definitely true.

Question 10

Syllogistic inference about carpeting and color: All offices on the 9th floor have wall-to-wall carpeting, and no wall-to-wall carpeting is pink; decide whether the conclusion "None of the 9th-floor offices has pink wall-to-wall carpeting" must foll…

Accepted Answer

Given data Premise 1: All 9th-floor offices have wall-to-wall carpeting. Premise 2: No wall-to-wall carpeting is pink. Conclusion to test: No 9th-floor office has pink wall-to-wall carpeting. Concept / Approach Use categorical logic: If all items in Group A have Property C, and nothing with Property C can be pink, then nothing in Group A has pink carpeting. Step-by-step reasoning Step 1: Every 9th-floor office ∈ {carpeted}.Step 2: {carpeted} ∩ {pink} = ∅ (by Premise 2, carpets cannot be pink).Step 3: Therefore, for any 9th-floor office, "pink wall-to-wall carpeting" is impossible. Verification / Alternative view Contrapositive thinking: If an office had pink wall-to-wall carpeting, then there would exist pink wall-to-wall carpeting, contradicting Premise 2. Common pitfalls Assuming some 9th-floor offices might lack carpets; Premise 1 rules that out. Final Answer Definitely true.

Question 11

Enrollment comparison puzzle: Class A has a higher enrollment than Class B, and Class C has a lower enrollment than Class B; assess whether the statement "Class A has a lower enrollment than Class C" must be true given only these two premises.

Accepted Answer

Given data Premise 1: A > B (Class A has higher enrollment than Class B). Premise 2: C < B (Class C has lower enrollment than Class B). Claim to test: A < C. Concept / Approach Chain inequalities around B and derive the relative order between A and C. Step-by-step reasoning Step 1: From A > B and C < B, we have A on the right of B and C on the left of B on the number line.Step 2: Therefore A > B > C, which implies A > C.Step 3: The claim A < C contradicts A > C derived above, so it cannot be true under any values consistent with the premises. Verification / Alternative view Example: Let B=30, A=35 (A>B), C=20 (CC. No choice of numbers that respects the premises will make A Common pitfalls Thinking "both are different from B" leaves A and C incomparable. Here, the directions relative to B determine a strict order. Final Answer Definitely false.

Question 12

A fruit basket contains more apples than lemons. There are more lemons in the basket than there are oranges. The basket contains more apples than oranges. If the first two statements are true, the third statement is

Accepted Answer

There are fewer oranges than either apples or lemons, so the statement is true. Easy method: (Try this method to solve without any confusion) 1. A fruit basket contains more apples than lemons = App > Lem 2. There are more lemons in the basket than there are oranges = Lem > Org Now, Combine the above two results: App > Lem > Org 3. The basket contains more apples than oranges (App > ... > Org) = Yes. Therefore, the given 3rd statement is true.

Question 13

The Shop and Save Grocery is south of Greenwood Pharmacy. Rebecca's house is northeast of Greenwood Pharmacy. Rebecca's house is west of the Shop and Save Grocery. If the first two statements are true, the third statement is

Accepted Answer

Because the first two statements are true, Rebecca's house is also northeast of the Shop and Save Grocery, which means that the third statement is false.

Question 14

Joe is younger than Kathy. Mark was born after Joe. Kathy is older than Mark. If the first two statements are true, the third statement is

Accepted Answer

Joe is younger than Kathy and older than Mark, so Mark must be younger than Kathy.

Question 15

On the day the Barton triplets are born, Jenna weighs more than Jason. Jason weighs less than Jasmine. Of the three babies, Jasmine weighs the most. If the first two statements are true, the third statement is

Accepted Answer

We only know that Jasmine weighs more than Jason. There is no way to tell whether Jasmine also weighs more than Jenna.

Question 16

The temperature on Monday was lower than on Tuesday. The temperature on Wednesday was lower than on Tuesday. The temperature on Monday was higher than on Wednesday If the first two statements are true, the third statement is

Accepted Answer

We know from the first two statements that Tuesday had the highest temperature, but we cannot know whether Monday's temperature was higher than Tuesday's

Question 17

Oat cereal has more fiber than corn cereal but less fiber than bran cereal. Corn cereal has more fiber than rice cereal but less fiber than wheat cereal. Of the three kinds of cereal, rice cereal has the least amount of fiber. If the first two state…

Accepted Answer

From the first statement, we know that bran cereal has more fiber than both oat cereal and corn cereal. From the second statement, we know that rice cereal has less fiber than both corn and wheat cereals. Therefore, rice cereal has the least amount of fiber.

Question 18

Martina is sitting in the desk behind Jerome. Jerome is sitting in the desk behind Bryant. Bryant is sitting in the desk behind Martina. If the first two statements are true, the third statement is

Accepted Answer

Given the information in the first two statements, Bryant is sitting in front of both Jerome and Martina, so the third statement must be false.

Question 19

Battery X lasts longer than Battery Y. Battery Y doesn't last as long as Battery Z. Battery Z lasts longer than Battery X. If the first two statements are true, the third statement is

Accepted Answer

The first two statements indicate that Battery Y lasts the least amount of time, but it cannot be determined if Battery Z lasts longer than Battery X.

Question 20

Spot is bigger than King and smaller than Sugar. Ralph is smaller than Sugar and bigger than Spot. King is bigger than Ralph. If the first two statements are true, the third statement is

Accepted Answer

Spot is bigger than King, and Ralph is bigger than Spot. Therefore, King must be smaller than Ralph.

Logical Problems Questions