% Q1 %✅ The preimage ℎ⁻¹(𝑦) ⊆ 𝑋 is the set of all states consistent with observation 𝑦 %✅ Smaller preimages correspond to lower state uncertainty %❌ If two different states have the same observation, the sensor must be noisy %❌ If two different states have the same observation, the sensor is not repeatable %✅ If ℎ is bijective, every preimage is a singleton %❌ All preimages have the same cardinality %❌ Preimages are always connected sets %✅ Preimages partition the state space

% Q2 %✅ All states differing only in 𝜃 belong to the same preimage %✅ The sensor cannot distinguish robot orientation %✅ The preimage has dimension at least 1 %❌ The sensor uniquely determines the full state %❌ The sensor is bijective %✅ The partition induced by ℎ is invariant under changes in 𝜃 %❌ The observation space must include 𝜃

% Q3 %❌ Dominance implies both sensors are bijective. %✅ ℎ₁ dominates ℎ₂ if every preimage of ℎ₁ is contained in a preimage of ℎ₂. %✅ If ℎ₁ dominates ℎ₂, then ℎ₂ = 𝑔 ∘ ℎ₁ for some function 𝑔. %❌ Dominance will depend on accuracy of both sensors. %✅ Dominance compares information content, not physical sensor quality. %❌ Dominance requires equal observation spaces. %✅ Dominance defines a partial order on sensors. %❌ All pairs of sensors are comparable under dominance.

% Q4 %❌ Both ℎ₁ and ℎ₂ must have identical numerical outputs. %❌ Both ℎ₁ and ℎ₂ must have the same observation space. %❌ Both ℎ₁ and ℎ₂ must be bijective. %✅ ℎ₁ and ℎ₂ induce the same partition of 𝑋. %✅ Each sensor dominates the other. %✅ ℎ₁ and ℎ₂ carry the same information about the state. %❌ The sensors must be physically identical. %❌ Both ℎ₁ and ℎ₂ have singleton preimages.

%Q5 %❌ Time-dependent sensors are always bijective. %❌ Time dependence ialways mproves sensor accuracy. %✅If ℎ(𝑥, 𝑡) = ℎ(𝑥, 𝑡′) for all 𝑡, 𝑡′ ∈ 𝑇, then there is no information about when the state was sensed. %✅The sensor ℎ(𝑥, 𝑡) = 𝑥 has one-dimensional preimages. %❌ Time must appear explicitly in the observation space 𝑌. %❌ The sensor ℎ(𝑥, 𝑡) = 𝑡 provides the state 𝑥 at time 𝑡. %❌ Time dependence eliminates state uncertainty. %❌ Time-dependent sensors must be absolute sensors.

%Q6 %❌ Fusion always produces a bijective sensor. %✅ The fused preimage equals the intersection of individual preimages. %✅ Fusion can reduce or preserve uncertainty. %❌ Fusion can increase uncertainty relative to both sensors. %✅ Fusion refines the partition of the state space. %❌ Fusion requires identical physical sensors. %❌ Fusion guarantees a unique state estimate. %❌ Fusion requires identical observation spaces.

%Q7 %✅ The set forms a lattice under refinement. %✅ The coarsest partition corresponds to a constant sensor. %✅ The finest partition corresponds to perfect sensing. %❌ All sensors are totally ordered. %❌ Every sensor is comparable to every other. %✅ Fusion corresponds to partition refinement. %✅Dominance corresponds to partition refinement. %❌ The sensor that yields the finest partition is always unique. %❌ The sensor that yields the coarsest partition is always unique.

%Which statements correctly summarize the preimage-based view of sensing? %✅ Sensors partition the state space %✅ Better sensors induce finer partitions %✅ Fusion corresponds to intersecting partitions %✅ Dominance is defined via preimage inclusion %❌ Sensors must be numeric to be compared %❌ Noise is required to define preimages %❌ Calibration always changes dominance %❌ All sensors can be totally ordered

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