Prove that for any relation R(X,Y)
Select Select(R (X=x0) (Y=y0))) =
Select (R(X=x0) and Y=y0))
What is an equivalent form of (Select(R (X=x0) or (Y=y0)))?
To prove that the expression (Select(R(X=x0) (Y=y0))) = Select(R(X=x0) and Y=y0)), we need to demonstrate that both expressions yield the same result.
Let's break down the steps to prove this equality:
1. Start with the left-hand side: (Select(R(X=x0) (Y=y0))).
- This expression represents selecting all tuples from relation R where the value of X is equal to x0 and the value of Y is equal to y0.
2. Now, consider the right-hand side: Select(R(X=x0) and Y=y0)).
- This expression represents selecting all tuples from relation R where the value of X is equal to x0 AND Y is equal to y0.
To prove they are equivalent, we need to show that every tuple that satisfies the condition in the left-hand side expression also satisfies the condition in the right-hand side expression, and vice versa.
Let's go through the steps:
1. Assume a tuple (a,b) satisfies the condition in the left-hand side expression, i.e., X=a and Y=b.
- Since the tuple satisfies the condition X=a, it also satisfies the condition X=a AND Y=b.
- Therefore, this tuple also satisfies the condition in the right-hand side expression.
2. Assume a tuple (c,d) satisfies the condition in the right-hand side expression, i.e., X=c AND Y=d.
- Since the tuple satisfies the condition X=c AND Y=d, it satisfies the condition X=c.
- Therefore, this tuple also satisfies the condition in the left-hand side expression.
Based on the above steps, we can conclude that the left-hand side expression and the right-hand side expression are equivalent.
Now, let's move on to finding an equivalent form of (Select(R(X=x0) or (Y=y0))).
An equivalent form of (Select(R(X=x0) or (Y=y0))) can be represented as (Select(R(X=x0)) or Select(R(Y=y0))).
This equivalent form breaks down the original expression into two separate select operations. The first select operation selects all tuples from relation R where the value of X is equal to x0, and the second select operation selects all tuples from relation R where the value of Y is equal to y0. Finally, the two select operations are combined using the OR operator to yield the final result.