Step-by-step Solution:

The problem asks us to identify the relationship between D-erythrose and four other molecules P, Q, R, and S.

The given structure is D-erythrose: $D\text{-erythrose} = \,\,\, \begin{array}{c} CHO \\ | \\ H-C-OH \\ | \\ H-C-OH \\ | \\ CH_2OH \end{array}$

D-erythrose is an aldotetrose, meaning it has four carbon atoms and an aldehyde functional group. It has two chiral centers at C-2 and C-3.

Let's analyze the relationship for each molecule in Column-I.

1. Analysis of Molecule P: $P = \,\,\, \begin{array}{c} CHO \\ | \\ HO-C-H \\ | \\ HO-C-H \\ | \\ CH_2OH \end{array}$ To compare P with D-erythrose, we can look at the configuration of the chiral centers. In P, the positions of -H and -OH groups are swapped at both C-2 and C-3 compared to D-erythrose. This means that molecule P is the mirror image of D-erythrose. Molecules that are non-superimposable mirror images of each other are called enantiomers. Therefore, P is the enantiomer of D-erythrose. In Column-II, 'Enantiomer' is listed as number 2. So, P → 2.

2. Analysis of Molecule Q: $Q = \,\,\, \begin{array}{c} CHO \\ | \\ H-C-OH \\ | \\ HO-C-H \\ | \\ CH_2OH \end{array}$ Comparing Q with D-erythrose:

• The configuration at C-2 ($H-C-OH$) is the same as in D-erythrose.
• The configuration at C-3 ($HO-C-H$) is inverted compared to D-erythrose ($H-C-OH$). Q and D-erythrose are stereoisomers because they have the same molecular formula and connectivity, but a different spatial arrangement of atoms. They are not mirror images of each other because the configuration differs at only one of the two chiral centers. Stereoisomers that are not enantiomers are called diastereomers. Therefore, Q is a diastereomer of D-erythrose. In Column-II, 'Diastereomer' is listed as number 1. So, Q → 1.

3. Matching with Options: Based on our analysis, we have found:

• P → 2 (Enantiomer)
• Q → 1 (Diastereomer)

Let's examine the given options:

• A: P→2, Q→3, R→2, S→2
• B: P→3, Q→1, R→1, S→2
• C: P→2, Q→1, R→1, S→3
• D: P→2, Q→3, R→3, S→1

Only option C has the correct assignments for both P and Q (P→2, Q→1). This strongly suggests that C is the intended answer. Let's analyze R and S to confirm, keeping in mind there might be errors in the question.

4. Analysis of Molecule R: $R = \,\,\, \begin{array}{c} CH_2OH \\ | \\ C=O \\ | \\ H-C-OH \\ | \\ CH_2OH \end{array}$ D-erythrose is an aldose (contains an aldehyde group), while R is a ketose (contains a ketone group). Both have the molecular formula C₄H₈O₄. Isomers that have the same molecular formula but different functional groups are called functional isomers. Functional isomers belong to a broader class called constitutional (or structural) isomers, which have different connectivity of atoms. Therefore, R is a constitutional isomer of D-erythrose. In Column-II, 'Constitutional isomer' is listed as number 3. So, the correct relationship should be R → 3. However, option C assigns R to 1 (Diastereomer). This is chemically incorrect. Diastereomers must have the same connectivity.

5. Analysis of Molecule S: $S = \,\,\, \begin{array}{c} CHO \\ | \\ H-C-OH \\ | \\ CH_2 \\ | \\ CH_2OH \end{array}$ Let's determine the molecular formula of S. It has 4 carbons. The number of hydrogens is 1 (in CHO) + 1 (at C-2) + 2 (at C-3) + 3 (in CH₂OH) = 7. The number of oxygens is 1 (in CHO) + 1 (at C-2) + 1 (in CH₂OH) = 3. So, the molecular formula is C₄H₇O₃. The molecular formula of D-erythrose is C₄H₈O₄. Since S has a different molecular formula, it is not an isomer of D-erythrose. Therefore, none of the relationships (1, 2, or 3) apply. Option C assigns S to 3 (Constitutional isomer). This is incorrect. The structure for S is likely a typo in the question paper.

Conclusion: Our rigorous analysis gives the mapping: P→2, Q→1, R→3, and S is not an isomer. Comparing this with the options, option C (P→2, Q→1, R→1, S→3) correctly identifies the relationships for P and Q. The relationships for R and S in option C are incorrect based on their given structures and standard chemical definitions. Given that in a multiple-choice test one must choose the best-fitting option, and the P→2 and Q→1 pairing is unique to option C, we select C as the answer, acknowledging the flaws in the question regarding R and S.