Discrete Mathematics

(Database)

(Part 2)

IT Page 2 : : Database by Yang-Sae Moon2Page 3 (1/7) 1 : Research .

Database by Yang-Sae Moon3Page 4 (2/7) 2 : Stafford , , , .

Database by Yang-Sae Moon4Page 5 (3/7)

3 : 5 .

123123Database by Yang-Sae Moon5Page 6 (4/7)

4: Smith (worker) Smith (manager) .

Database by Yang-Sae Moon6Page 7 (5/7)

5: .( COUNT .)

Database by Yang-Sae Moon7Page 8 (6/7)

6: .

Database by Yang-Sae Moon8Page 9 (7/7)

7: (manager) .

Database by Yang-Sae Moon9Page 10 : : (tuple relational calculus) Database by Yang-Sae Moon10Page 11 (1/2) (Relational Calculus) SQL (tuple relational calculus) (domain relational calculus) (declarative) , (expressive power) Database by Yang-Sae Moon11Page 12 (2/2) (relationally completeness) L , L ( ) L (relationally complete) . (), (aggregate functions), (grouping), (ordering) .

Database by Yang-Sae Moon12Page 13 (1/2) (range) . : $50,000 . {t | EMPLOYEE(t) and t.SALARY > 50000} , EMPLOYEE(t) t EMPLOYEE . t t.SALARY > 50000 . t . Database by Yang-Sae Moon13Page 14 (2/2) t . {t.FNAME, t.LNAME | EMPLOYEE(t) and t.SALARY > 50000} SQL . ( ) SELECTT.FNAME, T.LNAMEFROMEMPLOYEE TWHERET.SALARY > 50000;

Database by Yang-Sae Moon14Page 15 (1/2) {t1.A1, t2.A2, ..., tn.An | COND(t1, t2, ..., tn, tn+1, tn+2, ..., tn+m)}t1, t2, ..., tn, tn+1, tn+2, ..., tn+m Ai ti COND (formula)(formula) (atom) Ri(ti) ti Ri (: EMPLOYEE(t))(ti.A op tj.B), op (=, 50000) (true) (false) , {t.FNAME, t.LNAME | EMPLOYEE(t) and t.SALARY > 50000}Database by Yang-Sae Moon15Page 16 (2/2)(formula): and, or, not .F1 F2 (F1 and F2), (F1 or F2), not(F1), not(F2) .

: EMPLOYEE(t) and t.SALARY > 50000 Database by Yang-Sae Moon16Page 17 (quantifiers) (universal quantifier) () (for all ) (existential quantifier) () (their exists )(free) (bound) ( ) t (t) (t) , t , . p. 186 ( ):F1: d.DNAME = ResearchF2: (t)(d.DNUMBER = t.DNO) d F1 F2 t F2

Database by Yang-Sae Moon17Page 18 F , (t)(F) .F t F , (t)(F) , .

F , (t)(F) .F t F (t)(F) , .

F t (t)(F) , . F (t)(F) , . Database by Yang-Sae Moon18Page 19 (1/4) 1: Research .Q1: {t.FNAME, t.LNAME, t.ADDRESS | EMPLOYEE(t) and (d) (DEPARTMENT(d) and d.DNAME = Research and d.DNUMBER = t.DNO)} ( | ) . ( | ) such that EMPLOYEE(t), DEPARTMENT(d) t d .d.DNAME = Research (selection condition)( SELECT )d.DNUMBER = t.DNO (join condition)( EQUI-JOIN ) Database by Yang-Sae Moon19Page 20 (2/4) 2: Stafford , , , , .Q2: {p.PNUMBER, p.DNUM, m.LNAME, m.BDATE, m.ADDRESS | PROJECT(p) and EMPLOYEE(m) and p.PLOCATION = Stafford' and ((d)(DEPARTMENT(d) and p.DNUM = d.DNUMBER and d.MGRSSN = m.SSN))}

8: , , .Q8: {e.FNAME, e.LNAME, s.FNAME, s.LNAME | EMPLOYEE(e) and EMPLOYEE(s) and e.SUPERSSN = s.SSN} Database by Yang-Sae Moon20Page 21 (3/4) 3: 5 .

Q3': {e.LNAME, e.FNAME | EMPLOYEE(e) and ((x) (w) (PROJECT(x) and WORKS_ON(w) and x.DNUM = 5 and w.ESSN = e.SSN and x.PNUMBER = w.PNO))}

Database by Yang-Sae Moon21Page 22 (4/4) 4: Smith . - Q4: {p.PNAME | PROJECT(p) and (((e)(w)(EMPLOYEE(e) and WORKS_ON(w) and w.PNO = p.PNUMBER and e.LNAME = Smith and e.SSN = w.ESSN)) or((m)(d)(EMPLOYEE(m) and DEPARTMENT(d) and p.DNUM = d.DNUMBER and d.MGRSSN = m.SSN and m.LNAME = Smith)))}AND/OR/NOT UNION or INTERSECTION and not Database by Yang-Sae Moon22Page 23 -- (x) (P(x)) (notx) (not(P(x)))(x) (P(x)) not(x) (not(P(x)))(x) (P(x) and Q(x)) (notx) (not(P(x)) or not(Q(x)))(x) (P(x) or Q(x)) (notx) (not(P(x)) and not(Q(x)))(x) (P(x) or Q(x)) not(x) (not(P(x)) and not(Q(x)))(x) (P(x) and Q(x)) not(x) (not(P(x)) or not(Q(x))) ( (implies) )(x) (P(x)) (x) (P(x)) (notx) (P(x)) not(x) (P(x)) Database by Yang-Sae Moon23Page 24 (1/3) -- , 3 3: 5 .Q3: {e.LNAME, e.FNAME | EMPLOYEE(e) and((x) (not (PROJECT(x)) or (not (x.DNUM = 5) or((w) (WORKS_ON(w) and w.ESSN = e.SSN and x.PNUMBER = w.PNO)))))}

Q3 Q3: {e.LNAME, e.FNAME | EMPLOYEE(e) and F}F = (x) (not(PROJECT(x)) or F1)F1 = (not(x.DNUM = 5) or F2)F2 = (w) (WORKS_ON(w) and w.ESSN = e.SSN and x.PNUMBER = w.PNO) Database by Yang-Sae Moon24Page 25 (2/3) -- Q3 1.Q3 e 5 . .2.F, not(PROJECT(x)) PROJECT x .3.F1, not(x.DNUM = 5) PROJECT , DNUM 5 x .4.F2 , 5 PROJECT .

Q3: {e.LNAME, e.FNAME | EMPLOYEE(e) and F}F = (x) (not(PROJECT(x)) or F1)F1 = (not(x.DNUM = 5) or F2)F2 = (w) (WORKS_ON(w) and w.ESSN = e.SSN and x.PNUMBER = w.PNO) Database by Yang-Sae Moon25Page 26 (3/3) -- 6: .Q6: {e.FNAME, e.LNAME | EMPLOYEE(e) and (not(d) (DEPENDENT(d) and e.SSN = d.ESSN))} Q6 Q6 Q6: {e.FNAME, e.LNAME | EMPLOYEE(e) and ((d) (not(DEPENDENT(d)) or not(e.SSN = d.ESSN)))}

7: .Q7: {e.FNAME, e.LNAME | EMPLOYEE(e) and ((d) (p) (DEPARTMENT(d) and DEPENDENT(p) and e.SSN = e.MGRSSN and p.ESSN = e.SSN))} Database by Yang-Sae Moon26Page 27 (Safe Expression) (finite) (infinite) , : {t | not(EMPLOYEE(t))} EMPLOYEE ,

Database by Yang-Sae Moon27Page 28 : : Database by Yang-Sae Moon28Page 29 (1/2) (domain variables) ? , n n Database by Yang-Sae Moon29Page 30 (2/2): 0: John B. Smith .Q0: {uv | (q) (r) (s) (EMPLOYEE(qrstuvwxyz) and q = John and r = B and s = Smith)}EMPLOYEE : qrstuvwxyzBDATE u, ADDRESS v q(FNAME), r(MINIT), s(LNAME) (q, r, s) (QBE ): Q0: {uv | EMPLOYEE(John, B,Smith,t,u,v,w,x,y,z)} Database by Yang-Sae Moon30Page 31 (1/2) 1: Research .Q1: {qsv | (z) (EMPLOYEE(qrstuvwxyz) and (l) (m) (DEPARTMENT(lmno) and l = Research and m = z))}(m = z) (l = Research)

2: Stafford , , , .Q2: {iksuv | (j) (PROJECT(hijk) and (t) (EMPLOYEE(qrstuvwxyz) and (m) (n) (DEPARTMENT(lmno) and k = m and n = t and j = Stafford)))} Database by Yang-Sae Moon31Page 32 (2/2) -- 6: .Q6: {qs | (t) (EMPLOYEE(qrstuvwxyz) and (not (l) (DEPENDENT(lmno) and t = l)))}

7: .Q7: {sq | (t) (EMPLOYEE(qrstuvwxyz) and ((j) (DEPARTMENT(hijk) and ((l) (DEPENDENT(lmno) and t = j and l = t)))))} Database by Yang-Sae Moon32Page 33 (SELECT), (PROJECT), (UNION), (SET DIFFERNECE), (Cartesion product) , , - ( ) Database by Yang-Sae Moon33