The symbol → indicates functional dependency in the context of a relational…

Summary of which statements are true and why:

• (𝑋, 𝑌) → (𝑍, 𝑊) implies (𝑋, 𝑌) → 𝑍.

• Reason: Decomposition (projection) of functional dependencies allows splitting a right-hand side into individual attributes, so (X,Y) → Z (and (X,Y) → W).

• ((𝑋, 𝑌) → 𝑍 and 𝑊 → 𝑌) implies (𝑋, 𝑊) → 𝑍.

• Reason: From W → Y, augment with X to get XW → XY, then apply XY → Z to conclude XW → Z (augmentation + transitivity).

• (𝑋 → 𝑌 and 𝑌 → 𝑍) implies 𝑋 → 𝑍.

• Reason: This is the transitivity rule for functional dependencies.

• (𝑋, 𝑌) → (𝑍, 𝑊) implies 𝑋 → (𝑍, 𝑊).

• Reason: False in general. You cannot remove Y from the left side unless Y is already determined by X. A quick counterexample is a relation where two tuples share the same X but have different Y and different Z (or W) values — then X alone does not determine Z and W.