c) {X->Y, X->W, WY->Z} |= {X->Z} d) {XY->Z, Y->W} |= {XW->Z} e) {X->Z, Y->Z} |=

Question

c) {X->Y, X->W, WY->Z} |= {X->Z}

d) {XY->Z, Y->W} |= {XW->Z}

e) {X->Z, Y->Z} |= {X->Y}

f) {X->Y, XY->Z} |= {X->Z}

Question: Prove or disprove the following inference rules for functional dependencies. A proof can be made either by a proof argument or by using inference rules IR1 through IR3. A disproof should be done by demonstrating a relation instance that satisfies the conditions and functional dependencies in the left hand side of the inference rule but do not satisfy the conditions or dependencies in the right hand side: a) {W - > Y, X - > Z} |= {WX - > Y} b) {X - > Y} and Z subeq Y|= {X - > Z} c) {X - > Y, X - > W, WY - > Z} |= {X - > Z} d) {XY - > Z, Y - > W} |= {XW - > Z} e) {X - > Z, Y - > Z} |= {X - > Y} f) {X - > Y, XY - > Z} |= {X - > Z}

Explanation / Answer

(d) Disproof: X Y Z W

Let t 1 =x 1 y 1 z 1 w 1 and t 2 =x 1 y 2 z 2 w 1

The above two tuples satisfy XY ->Z and Y ->W but do not satisfy XW ->Z

(e) Disproof: X Y Z

Let t 1 =x 1 y 1 z 1 and t 2 =x 1 y 2 z 1

The above two tuples satisfy X ->Z and Y ->Z but do not satisfy X ->Y

(f)

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