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For this reason [0] = {y ∈ Z | zero ~ y} or [0] = {y ∈ Z | (0, y) ∈ R3} or [0] = {x ∈ Z | zero – y = okay. three , for y ∈ Z} i. e. , [0] = {{0 ± three, ± 6, ± 9,... } equally, [1] = {y ∈ Z | (1, y) ∈ R3} [1] = {... , – five, – 2, 1, four, 7, ... } and [2] = {... , – four, – 1, 2, five, eight, ... }. we'd like now not pass extra, simply because [3] = [0], [4], = [1], [5] = [2] ... We notice that if a, b, ∈ [0], then (a, b) ∈ R3, if e, f ∈ [1] the (e, f ) ∈ R3 and if x, y ∈ [2], then (x, y) ∈ R3. We additionally become aware of that [0] ∪ [1] ∪ [2] = Z and [0] ∩ [1] ∩ [2] = φ. hence the gathering Z3 = {[0], [1], [2]} is the partition of Z triggered by way of R3. This set Z3 = {[0], [1], [2]} is named the set of integers modulo three. three. four PARTIAL ORDER RELATION There are sorts of relatives which frequently come up in arithmetic: partial order relation which we need to examine and equivalence relation on which we have now commented. Definition three. four. 1. allow A be non-empty set. The relation R, the set of ordered pairs (a, b) ∈ AXA, is termed partial order relation within the set A if R satisfies the next stipulations: (1) Reflexive estate: (a, a) ∈ R for all a ∈ A. (2) Anti-symmetric estate: (a, b) ∈ R and (b, a) ∈ R, if and provided that a = b. (3) Transitive estate: If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R. A non-empty set A within which there's outlined a partial order relation is named ordered set. If it additionally occurs that for all a, b, ∈ A we both have (a, b) ∈ R or (b, a) ∈ R or a = b then R is named overall order relation on A. We become aware of that if a, b are unique components of A, then it isn't worthy that both (a, b) ∈ R or (b, a) ∈ R for each relation R. Definition three. four. 2. parts a and b are acknowledged to be similar with appreciate to a relation R, if both (a, b) ∈ R or (b, a) ∈ R or a = b. therefore, R is overall order relation on A if both (a, b) ∈ R or (b, a) ∈ R or a = b, for each pair a, b ∈ A. instance three. four. 1. The relation ‘‘>’’ outlined on N = {1, 2, three, ... }. (1) Reflexive a ( a for instance 1 >| 1 fake RELATION AND services fifty three (2) Anti-symmetric a > b and b >| a fake for instance 2 > 1 and 1 >| 2 (3) Transitive a > b and b > c real ⇒a>c because the relation isn't really reflexive, it's neither equivalence relation nor partial order relation yet ‘‘>’’ is a complete order relation. instance three. four. 2. The relation “<” on A = {1, 2, three, four, 5}. ≤ = {(x, y) | x ≤ y} i. e. , ≤ = {1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)} (1) Reflexive: for all a ∈ A, a = a, that's (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), ∈ R precise (2) Anti-symmetric (a, b) ∈ R and (b, a) ∈ R, if and provided that a = b real (3) Transitive: if (a, b) ∈ R, then (b, c) ∈ R, then (a, c) ∈ R, (2, three) ∈ R and (3, four) ∈ R, then (2, four) ∈ R actual for that reason ≤ is a partial order relation. instance three. four. three. enable A be any set: then the relation ‘‘⊆’’ outlined on P(A), the ability set of A. (1) Reflexive estate: A = A or A ⊆ a real (2) Anti-symmetric if A ⊆ B and B ⊆ A then A = B real (3) Transitive: if A ⊆ B and B ⊆ C, then A ⊆ C precise accordingly relation ⊆ outlined on P(A) is a partial order relation.

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