Given relations P(X,Y,Z), Q(X,Y,T), R(Y,V). P has tuples (X1,Y1,Z1),(X1,Y1,Z2),(X2,Y2,Z2),(X2,Y4,Z4)

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GATE-CS is a competitive examination conducted in India. This question is from the 2019 GATE-CS Database Management Systems paper.

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Question

Given relations P(X,Y,Z), Q(X,Y,T), R(Y,V). P has tuples (X1,Y1,Z1),(X1,Y1,Z2),(X2,Y2,Z2),(X2,Y4,Z4). Q has (X2,Y1,2),(X1,Y2,5),(X1,Y1,6),(X3,Y3,1). R has (Y1,V1),(Y3,V2),(Y2,V3),(Y2,V2). How many tuples does πₓ(σ(P.Y=R.Y ∧ R.V=V2)(P×R)) - πₓ(σ(Q.Y=R.Y ∧ Q.T>2)(Q×R)) return?

Accepted Answer

Answer: 1Concept: Relational Algebra — Set Difference and SelectionFirst expression: πₓ(σ(P.Y=R.Y ∧ R.V=V2)(P×R))R tuples with V=V2: (Y3,V2) and (Y2,V2). Match P.Y with these:P.Y=Y2 with (Y2,V2): P rows with Y=Y2: (X2,Y2,Z2). Projects to X={X2}.P.Y=Y3: No P rows with Y=Y3.First result: {X2}.Second expression: πₓ(σ(Q.Y=R.Y ∧ Q.T>2)(Q×R))Q rows with T>2: (X1,Y2,5), (X1,Y1,6), (X3,Y3,1 — T=1, not >2). So qualifying Q: (X1,Y2,5) and (X1,Y1,6).(X1,Y2,5): Q.Y=Y2. R rows with Y=Y2: (Y2,V3),(Y2,V2). Match. Contributes X={X1}.(X1,Y1,6): Q.Y=Y1. R rows with Y=Y1: (Y1,V1). Contributes X={X1}.Second result: {X1}.Set difference: {X2} - {X1} = {X2}. Answer = 1 tuple.Definition: Relational algebra operators: σ (selection) picks rows satisfying a condition; π (projection) extracts columns; × (Cartesian product); − (set difference).

Given relations P(X,Y,Z), Q(X,Y,T), R(Y,V). P has tuples (X1,Y1,Z1),(X1,Y1,Z2),(X2,Y2,Z2),(X2,Y4,Z4). Q has (X2,Y1,2),(X1,Y2,5),(X1,Y1,6),(X3,Y3,1). R has (Y1,V1),(Y3,V2),(Y2,V3),(Y2,V2). How many tuples does πₓ(σ(P.Y=R.Y ∧ R.V=V2)(P×R)) - πₓ(σ(Q.Y=R.Y ∧ Q.T>2)(Q×R)) return?

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Given relations P(X,Y,Z), Q(X,Y,T), R(Y,V). P has tuples (X1,Y1,Z1),(X1,Y1,Z2),(X2,Y2,Z2),(X2,Y4,Z4). Q has (X2,Y1,2),(X1,Y2,5),(X1,Y1,6),(X3,Y3,1). R has (Y1,V1),(Y3,V2),(Y2,V3),(Y2,V2). How many tuples does πₓ(σ(P.Y=R.Y ∧ R.V=V2)(P×R)) - πₓ(σ(Q.Y=R.Y ∧ Q.T>2)(Q×R)) return?

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