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Relations

Relations. Important Definitions . We covered all of these definitions on the board on Monday, November 7 th . Definition 1 Definition 2 Definition 3 Definition 4 Definition 5. Remember . A relation can be Symmetric and not antisymmetric R = { (1,3), (3,1)} on the set S = {1,2,3}

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Relations

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  1. Relations

  2. Important Definitions We covered all of these definitions on the board on Monday, November 7th. • Definition 1 • Definition 2 • Definition 3 • Definition 4 • Definition 5

  3. Remember • A relation can be • Symmetric and not antisymmetric • R = { (1,3), (3,1)} on the set S = {1,2,3} • Antisymmetric and not symmetric • R = { (1,1) (1,2) } on the set S = {1,2} • Symmetric and antisymmetric • Equality relation • Neither symmetric nor antisymmetric • R= {1,2), (2,1), (1,3)} on the set S = (1,2,3}

  4. Remember • A relation can be • Reflexive • R = { (1,1), (2,2), (3,3)} on the set S = {1,2,3} • Irreflexive • R = { (1,2), (2,3)} • Neither reflexive nor irreflexive • R1 = { (1,2), (2,1), (1,1)} on the set S= {1,2} • R2 = { (2,2) }

  5. Combining Relations • Since relations from A to B are subsets of A X B, two relations from A to B can be combined in any way two sets can be combined. • Union • Intersection • Set difference

  6. Composite Relations • Let R be a relation from a set A to a set B and Let S be a relation from set B to set C. • The composite of R and S is the relation consisting of the ordered pairs (a,c) where a A, c  C, and for which there exists an element b  B such that (a,b)  R & (b,c)  S • We denoted the composite of R and S by • S ° R.

  7. the Powers of a relation • The powers of a relation R can be recursively defined from the definition of a composite of two relations. • Let R be a relation on the set A. • The powers Rn, n = 1,2,3,…, are defined recursively by • R1 = R and Rn+1 = Rn°R • THEOREM: The relation R on a set A is transitive iff Rn⊆R for n = 1,2,3, …

  8. n-ARY RELATIONS • Relations among elements of more than two sets often arise. • Examples • Student: name, major, gpa • Airline flight: number, origin, destination, departure time, arrival time • Points on a line • Let A1, A2, …,An be sets. • An n-ary relation on these sets is a subset of A1 X A2X… X An

  9. Relational Database Theory • The sets A1, A2, …,An are called the domains of the relation, and n is called its degree. • The relational data model is based on the concept of a relation. • A database consists of records which are n-tuples made up of fields. • Relations used to represent database are also called tables. • Each column of the table corresponds to an attribute of the database.

  10. Database Keys • A domain of an n-ary relation is called a primary key when the value of the n-tuple from this domain determines the n-tuple. • Combinations of domains can also uniquely identify n-tuples in an n-ary relation. When the values of a set of domains determin an n-tuple in a relation, the Cartesian product of these domains is called a composite key.

  11. Operations on n-ary relations • Select • Let R be an n-ary relation and C a condition that elements in R may satisfy. Then the selection operator sC maps the n-ary relation R to the n-ary relation of all n-tuples from R that satisfy the condition C. • Project • The projection Pi1, i2, …immaps the n-tuple (a1, a2, …,an) to the m-tuple (ai1,ai2, …aim), where m <= n.

  12. Operations on n-ary relations • Join • Let R be a relation of degree m and S a relation of degree n. The join Jp (R,S), where p <= m and p <= n, is a relation of degree m+n-p that consists of all (m+n-p)-tuples (a1,a2, …,am-p, c1,c2,…,cp, b1,b2,…,bn-p), where the m-tuple (a1,a2, …,am-p, c1,c2,…,cp)belongs to R and the n-tuple (c1,c2,…,cp,,b1,b2,…,bn-p) belongs to S.

  13. Representing relations using matrices • A relation between finite sets can be represented using a zero-one matrix • Suppose that R is a relations from A = {a1, a2, …, am} to B= {b1, b2, …, bn} • The relation R can be represented by the matrix MR = [mij]where mij= 1 if (ai,bj) R, mij= 0 if (ai,bj) ∉R

  14. Representing relations using digraphs • A directed graph, or digraph, consists of a set V of vertices (or nodes) together with a set E of ordered pairs of elements of V called edges (or arcs). • The vertex a is called the initial vertex of the edge (a,b), and the vertex b is called the terminal vertex of this edge.

  15. Partial Ordering • A binary relation on a set S that is reflexive, antisymmetric, and transitive is called a partial ordering on S. • Hasse diagram • If S is finite, we can visually depict a partially ordered set by using a Hasse diagram. Each of the elements of S is represented by a dot, called a node or vertex of the diagram. If x is an immediate predecessor of y, then the node for y is placed above the node for x and the two nodes are connected by a straight line segment. • Draw the Hasse diagram for the relation “x divides y” on the set {1,2,3,6,12,18} • PERT (program evaluation and review technique) chart is a Hasse diagram with time added

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