數(shù)據(jù)庫系統(tǒng)基礎(chǔ)教程第五章答案(參考版)
2025-06-29 20:37本頁面
【正文】 R. Therefore the mutative law for union holds.Exercise For bags:Suppose a tuple t occurs n and m times in bags R and S respectively. In the intersection of these two bags R ∩ S, tuple t would appear min( n,m ) times. Likewise in the intersection of these two bags S ∩ R, tuple t would appear min( m,n ) times. Both sides of the relation yield the same result. For sets:A tuple t can only appear at most one time. Tuple t might appear each in sets R and S one or zero times. The binations of number of occurrences for tuple t in R and S respectively are (0,0), (0,1), (1,0), and (1,1). Only when tuple t appears in at least one of the sets R and S will the intersection R ∩ S have the tuple t. The same reasoning holds when we take the intersection S ∩ R.Therefore the mutative law for intersection holds.Exercise For bags:Suppose a tuple t occurs n times in bag R and tuple u occurs m times in bag S. Suppose also that the two tuples t,u can successfully join. Then in the natural join of these two bags R S, the joined tuple would appear nm times. Likewise in the natural join of these two bags S R, the joined tuple would appear mn times. Both sides of the relation yield the same result. For sets:An arbitrary tuple t can only appear at most one time in any set. Tuples u,v might appear respectively in sets R and S one or zero times. The binations of number of occurrences for tuples u,v in R and S respectively are (0,0), (0,1), (1,0), and (1,1). Only when tuple u exists in R and tuple v exists in S will the natural join R S have the joined tuple. The same reasoning holds when we take the natural join S R.Therefore the mutative law for natural join holds.Exercise For bags:Suppose tuple t appears m times in R and n times in S. If we take the union of R and S first, we will get a relation where tuple t appears m + n times. Taking the projection of a list of attributes L will yield a resulting relation where the projected attributes from tuple t appear m + n times. If we take the projection of the attributes in list L first, then the projected attributes from tuple t would appear m times from R and n times from S. The union of these resulting relations would have the projected attributes of tuple t appear m + n times.For sets:An arbitrary tuple t can only appear at most one time in any set. Tuple t might appear in sets R and S one or zero times. The binations of number of occurrences for tuple t in R and S respectively are (0,0), (0,1), (1,0), and (1,1). Only when tuple t exists in R or S (or both R and S) will the projected attributes of tuple t appear in the result.Therefore the law holds.Exercise For bags:Suppose tuple t appears u times in R, v times in S and w times in T. On the left hand side, the intersection of S and T would produce a result where tuple t would appear min(v , w) times. With the addition of the union of R, the overall result would have u + min(v , w) copies of tuple t. On the right hand side, we would get a result of min(u + v, u + w) copies of tuple t. The expressions on both the left and right sides are equivalent.For sets:An arbitrary tuple t can only appear at most one time in any set. Tuple t might appear in sets R,S and T one or zero times. The binations of number of occurrences for tuple t in R, S and T respectively are (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0) and (1,1,1). Only when tuple t appears in R or in both S and T will the result have tuple t. Therefore the distributive law of union over intersection holds.Exercise Suppose that in relation R, u tuples satisfy condition C and v tuples satisfy condition D. Suppose also that w tuples satisfy both conditions C and D where w ≤ min(v , w). Then the left hand side wil
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