4.1 Product Sets and Partitions 1. Product Sets: Let A and B be two nonempty sets. The product set or Cartesian product A B is the set of all ordered pairs a,b with a ∈ A and b ∈ B, that is, A B a,b| a ∈ A and b ∈ B . Example: Let A 1,2,3 and B r,s. A\Br s A B 1,r,1,s,2,r,2,s,3,r,3,s. 1 1,r1,s |A B| |A||B| 2 2,r2,s 3 3,r3,s Example: Let A B R. Then A B R2. Cartesian Product:
A1 A2 Am a1,...,am | ak ∈ Ak , k 1,...,m
1 2. Partitions A partition or a quotient set of a nonempty set A is a collection P of nonempty subsets of A such that (1) Each element of A belongs to one of the sets of P.
(2) If A1 and A2 are distinct elements of P, then A1 ∩ A2 . The sets in P are called the blocks or cells of the partition. Example 6: Let A a,b,c,d,e,f,g,h. Consider subsets of A:
A1 a,b,c,d, A2 a,c,e,f,g,h, A3 a,c,e,g,
A4 b,d, A5 f,h
Then A1, A2 is not a partition because A1 ∩ A2 .
A1, A5 is not a partition because e,g ∉ A1 and A5.
A2, A4 and A3, A4, A5 are partitions of A.
2 4.2 Relations and Digraphs 1. Relations Let A and B be two non-empty sets. A relation R from A to B is a subset of A B a,b| a ∈ A, b ∈ B. If R ⊆ A B and a,b ∈ R, then we say that a is related to b by R, denoted as aRb. Other notation: a R b. If a is not related to b, then a ≠R b.
Example: Let A and B be two sets of real numbers. Define a relation R from A to B as
y aRbif and only if a b. 4 2
R x,x| x is a real number -4 -2 2 4 x -2
The graph of this relation: -4
Example: Let A ℤ. Define R from A to A by aRbif and only if a | b.
3 R a,qa| q is an integer 1,1, 1,2, ...,2,2,2,4, ...
y 12
10
8
Graph of R : 6
4
2
0 1 2 3 4 x
2. Domain and Range of a Relation The domain of a relation, denoted as Dom(R), is a subset of A containing all elements that are related to some elements in B. The range of a relation, denoted as Ran(R), is a subset of B containing all elements that are paired with some elements in A.
Example: A function is also a relation. The domain and range of a
4 function are the same the domain and range of that relation. Let fx x2 − 1 , gx e−x, hx lnx 3.
Example: Let A be the set of all real numbers. Consider the relation R: xRyif and only if x and y satisfy the equation x2 y2 4 9 1
3. The Matrix of a Relation The matrix of a given relation R is defined by
1 if ai,bj ∈ R M mij where mij 0 if ai,bj ∉ R Example: Sketch the direct graph of R if the matrix of R is given: M
5 4. The Directed Graph (Digraph) of a Relation Let R be a relation on A A A. Then the directed graph of R is a graph whose vertices are elements of A and which has an edge from vertex a to b if a,b in R. The in-degree of the element a in A is defined by the number b in A such that b,a ∈ R. The out-degree of the element a in A is defined by the number b in A such that a,b ∈ R.
6 4.3 Paths in Relations and Direct Graphs 1. Paths: A path of length n in R from a to b is a finite sequence: a,x1,x2,...,xn−1,b, beginning with a and ending with b such that aRx1, x1Rx2,...,xn−1Rb. Note: In a direct graph, a path is a succession of edges. A cycle is a path that begins and ends at the same vertex. Example 1: There are several paths:
P1: 1,2,3 of length 2; P2: 1,2,5,4,3 of length 4;
P3: 2,2 of length 1, is also a cycle;
P4: 1,2,5,1 of length 3, is also a cycle.
2. Relation Rn and R: Let A be a set and x and y be in A. Define the relation Rn as: xRny if there is a path of length n from x to y. Rnx y| y ∈ A, and yRnx
7 k n R x y| y ∈ A, and yR x for some k (all possible k) n≥1 R
Example 4:A 1,2,3,4,5,6, R: Figure 4.12, R2: Figure 4:13
Example 5:A a,b,c,d,e, R a,a, a,b, b,c, c,e, c,d, d,e 11000 00100 M 00011 , 00001 00000
8 2 11000 11100 00100 00011 M2 00011 00001 00001 00000 00000 00000 3 11000 11111 00100 00001 M3 00011 00000 00001 00000 00000 00000
9 4 11000 11112 00100 00000 M4 00011 00000 , 00001 00000 00000 00000 5 11000 11112 00100 00000 M5 00011 00000 00001 00000 00000 00000 M M2 M3 M4 M5
10 11000 11100 00100 00011 00011 00001 00001 00000 00000 00000 11111 11112 44323 00001 00000 00112 00000 00000 00012 00000 00000 00001 00000 00000 00000 R2 a,a,a,b,a,c,b,d,b,e,c,e R3 a,a,a,b,a,c,a,d,a,e R
11 3. Matrix Representing a Relation: (1) Boolean Matrix Operations (Section 1.5): A Boolean matrix is a matrix whose entries are 0 or 1.
Let A aij , and B bij be m n Boolean matrices.
(i) the join of A and B: C A ∨ B cij where
1 if aij 1 or bij 1 cij 0 if aij 0 and bij 0
(ii) the meet of A and B: D A ∧ B dij where
1 if aij 1 and bij 1 dij 0 if aij 0 or bij 0
(iii) Let A aij be m p, and B bij be p n Boolean matrices. the Boolean product of A and B: E A ⊙ B eij where
1 if aik 1 and bkj 1 for some k,1≤ k ≤ p eij 0 otherwise
12 110 1000 010 Example: A , B 0110 110 1011 001
1110 0110 A ⊙ B 1110 1011
(2) Matrix Representation of Rn:
Let R bearelationonafinitesetA a1,...,an , and let MR be the n n matrix representing R. Then MR2 MR ⊙ MR is the matrix 2 representing R and MRn MR ⊙ MR ⊙ MR is the matrix representing Rn. Example 5: A a,b,c,d,e, R a,a,a,b,b,c,c,e,c,d,d,e
13 11000 00100
MR 00011 , 00001 00000
11000 11000 11100 00100 00100 00011
MR ⊙ MR 00011 00011 00001 00001 00001 00000 00000 00000 00000 R2 a,a,a,b,a,c,b,d,b,e,c,e
4. Composition of Paths:
Let P1: a,x1,...,xn−1,b be a path in a relation R and P2: b,y1,y2,...,ym−1,c
14 be a path in R. Then the composition of P1 and P2 is a path P P1 ∘ P2: a,x1,...,xn−1,b,y1,...,ym−1,c of length n m from a to c. Example 7:
15 4.4 Properties of Relations: 1. Reflexive and Irreflexive Relations: A relation R onasetA is reflexive if a,a ∈ R for all a ∈ A. A relation if irreflexive if none of a,a is in R.
Note that the diagonal entries of MR are 1’s if R is reflexive and are all 0′s if R is irreflexive. Example 1:
(a) R1 a,a| a ∈ A - reflexive
(b) R2 a,b ∈ A A| a ≠ b - cannot be reflexive so it irreflexive
2. Symmetric, Asymmetric, and Antisymmetric Relations: A relation R is symmetric if b,a ∈ R whenever a,b ∈ R. A relation R is asymmetric if b,a ∉ R whenever a,b ∈ R. A relation R is antisymmetric if whenever a,b ∈ R and b,a ∈ R, then a b.
16 Example 2: Let A ℤ and R a,b ∈ A A| a b R is NOT symmetric since b ≰ a if a ≤ b. So it is asymmetric. R is not antisymmetric since a b, a ≠ b
3. Graph of a Symmetric Relation: The graph with undirected edges is called undirected graph or graph. The graph of a symmetric relation can be expressed by a undirected graph. Vertices a and b are called adjacent vertices and the matrix is called adjacent matrix.
4. Transitive Relations: A relation R onasetA is transitive if whenever a,b ∈ R and b,c ∈ R, then a,c ∈ R. Theorem 1: A relation R is transitive if and only if it satisfies the following property: If there is a path of length greater than 1 from a to b,then there is a path
17 of length 1 from a to b
5. Summary: Let R be a relation on a set of A. Reflexivity of R means that a,a ∈ R for all a ∈ A. Symmetry of R means that a,b ∈ R if and only if b,a ∈ R. Transitivity of R means that a,b ∈ R and b,c ∈ R implies a,c ∈ R.
18 4.5 Equivalence Relations A relation R onasetA is an equivalence relation if it is reflexive, symmetric and transitive. Example: a ≡ b (mod n) "a is congruent to b mod n" meaning a qn r and b pn r
Note: fna fnb or a − b q − pn, n|a − b Determine if R a,b | a ≡ b (mod 2) is an equivalence relation. Example: Let A ℤ and let n ∈ ℤ. Define R a,b ∈ A A| a ≡ b mod n n3 example: R a,b ∈ A A| a ≡ b mod 3 For m 0,1,2,..., 0,3m ∈ R, 1,3m 1 ∈ R, 2,3m 2 ∈ R Define R0 b ∈ A| b ≡ 0 mod 3 , R1 b ∈ A| b ≡ 1 mod 3 and R2 b ∈ A| b ≡ 2 mod 3 .
19 R0, R1 and R2 are called the equivalence classes of R, denoted as a, a 0,1,2. Notes: (a) P R0, R1, R2 forms a partition of A. (b) R0 R3. We say, Ra and Rb are distinct if and only if Ra ≠ Rb.
Theorem 2: Let R be an equivalent relation on A and P be a collection of all distinct relative sets Ra for all a in A. Then P is a partition of A, and R is the equivalence relation determined by P.
Quotient set: P A/R, is a special quotient set of A. Example: P R0, R1, R2 the quotient set of ℤ constructed from R.
20