StuckinMaths
New member
- Joined
- Mar 3, 2012
- Messages
- 5
∈≡ ≈ ≅ ≠ ≤ ≥ · ~ ± ∓ ∤ ◅∈ ∉ ⊆ ⊂ ∪ ∩ ⊥ ô
∫ Σ → ∞ Π Δ Φ Ψ Ω Γ ∮ ∇∂ √ ∅ °
α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ σ τ υ φ χ ψ ω
⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁿ ₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋
⅓ ⅔ ⅕ ⅖ ⅗ ⅘ ⅙ ⅚ ⅛ ⅜ ⅝ ⅞
This question concerns the two relations ~1 and ~2 which are defined on the integers Z by
x~1 y if 3x² - y² is divisible by 2,
x~2 y if 3x² - y² ≥ 0
a) for each of the two relations, state whether or not the relation has the reflexive, symmetric and transitive properties, giving a proof or counter example in each case. Hence determine which of the relations is an equivalence relation.
b) For the equivalence relation you found in part (a), show that [[3]] = {2k+1: k ∈ Z}
Thank you for your help!
∫ Σ → ∞ Π Δ Φ Ψ Ω Γ ∮ ∇∂ √ ∅ °
α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ σ τ υ φ χ ψ ω
⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁿ ₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋
⅓ ⅔ ⅕ ⅖ ⅗ ⅘ ⅙ ⅚ ⅛ ⅜ ⅝ ⅞
This question concerns the two relations ~1 and ~2 which are defined on the integers Z by
x~1 y if 3x² - y² is divisible by 2,
x~2 y if 3x² - y² ≥ 0
a) for each of the two relations, state whether or not the relation has the reflexive, symmetric and transitive properties, giving a proof or counter example in each case. Hence determine which of the relations is an equivalence relation.
b) For the equivalence relation you found in part (a), show that [[3]] = {2k+1: k ∈ Z}
Thank you for your help!