(Solved): Consider the Armstrongs Axioms, where X, Y, Z and W are sets of attributes: Question 7 (10 marks ...

Consider the Armstrong’s Axioms, where X, Y, Z and W are sets of attributes:

Question 7 (10 marks) Consider the Armstrong's Axioms, where X, Y, Z and W are sets of attributes: Reflexivity: If \( \mathrm{X} \supseteq \mathrm{Y} \), then \( \mathrm{X} \rightarrow \mathrm{Y} \) Augmentation: If \( \mathrm{X} \rightarrow \mathrm{Y} \), then \( \mathrm{XZ} \rightarrow \mathrm{YZ} \) Transitivity: If \( \mathrm{X} \rightarrow \mathrm{Y} \) and \( \mathrm{Y} \rightarrow \mathrm{Z} \), then \( \mathrm{X} \rightarrow \mathrm{Z} \) We can derive the following rules using Armstrong Axioms only: Union: If \( \mathrm{X} \rightarrow \mathrm{Y} \) and \( \mathrm{X} \rightarrow \mathrm{Z} \), then \( \mathrm{X} \rightarrow \mathrm{YZ} \) Decomposition: If \( \mathrm{X} \rightarrow \mathrm{YZ} \), then \( \mathrm{X} \rightarrow \mathrm{Y} \) and \( \mathrm{X} \rightarrow \mathrm{Z} \) Pseudo Transitivity: If \( \mathrm{X} \rightarrow \mathrm{Y} \) and \( \mathrm{WY} \rightarrow \mathrm{Z} \), then \( \mathrm{XW} \rightarrow \mathrm{Z} \) For example, the following sequence derives Decomposition: 1. \( Y Z \rightarrow Y \) (Reflexivity) 2. \( \mathrm{YZ} \rightarrow \mathrm{Z} \) (Reflexivity) 3. \( \mathrm{X} \rightarrow \mathrm{YZ} \) (given in "if" part of Decomposition) 4. \( \mathrm{X} \rightarrow \mathrm{Y}(1,3 \), Transitivity) 5. \( \mathrm{X} \rightarrow \mathrm{Z}(2,3 \), Transitivity) Steps 4 and 5 indicate that we have derived the FDs in the "then" part of Decomposition, so we are done. Note that each step in the above derivation is obtained from Armstrong Axioms (such as 1 and 2), or the FDs given in the "if" part of the rule to be derived (such as 3), or applying Armstrong Axioms to previously obtained FDs (such as 4 and 5). Other than these, you are not allowed to use anything else. You are asked to derive the Union rule and the Pseudo Transitivity rule, and justify each step as shown in the above example. 5 marks for deriving each rule.