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Various relations and sets, focusing on inequalities and equivalences. Topics include subset relations, cartesian products, reflexivity, symmetry, transitivity, and equivalence relations. It also covers the concepts of reflexive, symmetric, and transitive closures, and the difference between equivalence and congruence relations.
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(a, b) ∈ R
a
b
∀a ∀b ¬ (R (a, b) ∧ R (b, a))
a
b
a
b
a
b
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(b) 6 = ∅
a
{u, v, w, x}
b
{u, v, y, z}
c
{y, z, r, s}
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a
b
R (a, b) ≡ {x | x
a} ⊆ {x | x
b} ≡ ∀x, x
a =⇒ x
b
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b
R (a, b) ≡
(a)∩
(b) = ∅
a
b
R (a, b) ≡
(a)∩
(b) 6 = ∅
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R (a, b) ≡ ∃c, a ∈
(c) , b ∈
(c)
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{(a, b) | a
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{(a, b) | a
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{(a, b) | a
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x
x¯ = {y | y
x}
n ∈ N
{y | y
n
x
¯x = {y | y
x} = {y | y
x}
(m, d)
{y | y
(m, d)}
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