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離散數學

• Binary Relation on a Set

  • Definition: A binary relation R on a set A is a subset of A × A or a relation from A to A
  • Ex:A = {1,2,3,4}
    • R = {(a,b) | a divides b} are (1,1), (1, 2), (1,3), (1, 4), (2, 2), (2, 4), (3, 3), and(4, 4).

• Reflexive Relations

  • 如果element x ∊A 則(x,x)∊ R
  • EX:The following relations on the integers are reflexive:
    R1 = {(a,b) | a ≤ b},
    R3 = {(a,b) | a = b or a = −b},
    R4 = {(a,b) | a = b}.
    The following relations are not reflexive:
    R2 = {(a,b) | a > b} (note that 3 ≯3),
    R5 = {(a,b) | a = b + 1} (note that 3 ≠3 + 1),
    R6 = {(a,b) | a + b ≤ 3} (note that 4 + 4 ≰3)

• Symmetric Relations

  • 如果element (x,y)∊A 則 (y,x)∊A
  • Ex: The following relations on the integers are symmetric:
    R3 = {(a,b) | a = b or a = −b},
    R4 = {(a,b) | a = b},
    R6 = {(a,b) | a + b ≤ 3}.
    The following are not symmetric:
    R1 = {(a,b) | a ≤ b} (note that 3 ≤ 4, but 4 ≰ 3),
    R2 = {(a,b) | a > b} (note that 4 > 3, but 3 ≯ 4),
    R5 = {(a,b) | a = b + 1} (note that 4 = 3 + 1, but 3 ≠4 + 1).