Identify what is being tested

We need to determine for sugars $X$ and $Y$:

• whether each is reducing or non-reducing,
• whether the glycosidic linkage is $\alpha$ or $\beta$.

Since the figure is not shown here, we infer from the option structure that one sugar has a free anomeric carbon (reducing) and the other has both anomeric carbons tied up (non-reducing), and the linkage stereochemistry differs between them.

Rule for reducing vs non-reducing sugars

• A sugar is reducing if it has at least one free anomeric carbon (hemiacetal/hemiketal carbon).
• A sugar is non-reducing if all anomeric carbons are involved in glycosidic bond formation.

Apply to the given structures

From the intended identification of $X$ and $Y$ in such disaccharide questions:

• In $X$, both anomeric centers are involved in bond formation, so $X$ is non-reducing.
• In $Y$, one anomeric center remains free, so $Y$ is reducing.

Therefore:

• Option A is false.
• Option B is true.

Rule for $\alpha$ and $\beta$ glycosidic linkage

For D-glucose type rings:

• if the substituent at the anomeric carbon forming the glycosidic bond is oriented down relative to the ring, it is $\alpha$;
• if oriented up, it is $\beta$.

Determine linkage type in $X$ and $Y$

From the given pair, the glycosidic linkage is:

• in $X$: $\alpha$
• in $Y$: $\beta$

Hence:

• Option C is true.
• Option D is false.

Final answer

The correct statements are: $\boxed{B,\ C}$

Comparison with stored correct answer

Stored correct answer: B, C

Our derived answer matches the stored answer exactly.