http://www.flaticon.com/

https://www.ttnews.com/articles/autonomous-cars-may-cut-workplace-deaths-official-says

https://web.expasy.org/prolune/images/Prolune_3005_3.jpg

https://www.franchiseindia.com/brands/f-mart.29864

http://archive.jsonline.com/blogs/news/361521391.html

http://www.controlcommandescape.com/reviews/starcraft-2-review/https://www.windowscentral.com/blizzard-takes-swipe-pay-win-games-their-latest-starcraft-2-ad

https://web.expasy.org/prolune/images/Prolune_3005_3.jpg

5

2

3

1

4

5

2

3

1

4

𝐴

𝐵

1

𝐴

𝐵

1

𝑺𝒆𝒙

𝑷𝒂𝒖𝒍

𝑵𝒂𝒉𝒊𝒂

𝑰𝒔𝒂𝒃𝒆𝒍𝒍𝒆

𝑮𝒆𝒐𝒓𝒈

•

•

•

𝑺𝒆𝒙

𝑷𝒂𝒖𝒍

𝑵𝒂𝒉𝒊𝒂

𝑰𝒔𝒂𝒃𝒆𝒍𝒍𝒆

𝑮𝒆𝒐𝒓𝒈

{𝑃𝑎𝑢𝑙, 𝑁𝑎ℎ𝑖𝑎, 𝐼𝑠𝑎𝑏𝑙𝑙𝑒, 𝐺𝑒𝑜𝑟𝑔}

𝑆𝑒𝑥 ∶ 𝑂 → {𝑀, 𝐹}, 𝐴𝑔𝑒 ∶ 𝑂 → ℕ

𝑂

𝒔𝒉𝒂𝒑𝒆 ∶ 𝑂 → { , , , , }𝒄𝒐𝒍𝒐𝒓 ∶ 𝑂 → {white, black, gray}𝒙 ∶ 𝑂 → ℝ𝒚 ∶ 𝑂 → ℝ

𝑥

𝑦

𝑥

𝑦

𝒔𝒉𝒂𝒑𝒆 𝑿 =

•

•

𝑥

𝑦

𝒔𝒉𝒂𝒑𝒆 𝑿 𝒊𝒔 𝒄𝒐𝒏𝒗𝒆𝒙

𝑥

𝑦

𝒔𝒉𝒂𝒑𝒆 𝑿 = ∧ color(X) = gray

𝑥

𝑦

𝒙 𝑿 ∈ 𝟓, 𝟏𝟓 ∧ 𝒚 𝑿 = [𝟑, 𝟏𝟒]

𝑥

𝑦

𝒙 𝑿 + 𝒚 𝑿 ≥ 𝟏𝟔

𝑥

𝑦

In the neighborhood of 𝟒, 𝟒

𝒙 𝑿 − 𝟒 𝟐 + 𝒚 𝑿 − 𝟒 𝟐 ≤ 𝟏𝟔

×

𝑥

𝑦

In the neighborhood of within a radius of 4

𝑥

𝑦

𝒍𝒐𝒈 𝒙(𝑿)𝟑 + 𝒚(𝑿) − 𝟐 ⋅ 𝒙 𝑿 ⋅ 𝒚 𝑿 + 𝒚 𝑿 ≤𝟕𝟔𝟐𝟑. 𝟏

𝒆𝟐 ⋅ 𝒙(𝑿)𝟐.𝟑−𝒚(𝑿)

•

•

•

𝑨𝒍𝒍 𝒂𝒓𝒆 𝒃𝒍𝒂𝒄𝒌

𝑥

𝑦

𝐴

𝐵

1

𝟏 × ×

𝟐

𝟑 ×

𝟒 ×

𝟓 ×

𝟔 ×𝑻𝒓𝒖𝒆 𝒑𝒐𝒔𝒊𝒕𝒊𝒗𝒆 𝒓𝒂𝒕𝒆 – 𝒇𝒂𝒍𝒔𝒆 𝒑𝒐𝒔𝒊𝒕𝒊𝒗𝒆 𝒓𝒂𝒕𝒆 =

2

3−1

3=1

3

𝑀𝑊 𝑋 ∈ 128, 151 ∧

𝑛𝐴𝑇 𝑋 ∈ 20, 27 ∧

𝑛𝐶 𝑋 ∈ 9,12 →

𝑉𝑎𝑛𝑖𝑙𝑖𝑛

5

2

3

1

4

𝒔𝒉𝒂𝒑𝒆 𝑿 =𝒔𝒉𝒂𝒑𝒆 𝑿 =

𝒔𝒉𝒂𝒑𝒆 𝑿 =

𝒔𝒉𝒂𝒑𝒆 𝑿 =

𝒔𝒉𝒂𝒑𝒆 𝑿 𝒊𝒔 𝒂 𝒑𝒐𝒍𝒚𝒈𝒐𝒏

𝒔𝒉𝒂𝒑𝒆 𝑿 𝒉𝒂𝒔 𝟒 𝒔𝒊𝒅𝒆𝒔

𝒔𝒉𝒂𝒑𝒆 𝑿 𝒊𝒔 𝒂𝒏𝒚 𝒇𝒐𝒓𝒎

𝒔𝒉𝒂𝒑𝒆 𝑿 𝒊𝒔 𝒄𝒐𝒏𝒗𝒆𝒙

𝒔𝒉𝒂𝒑𝒆 𝑿 =

𝑥

𝑦

⟹

𝒔𝒉𝒂𝒑𝒆 𝑿 = 𝒔𝒉𝒂𝒑𝒆 𝑿 =

𝒔𝒉𝒂𝒑𝒆 𝑿 =

𝒔𝒉𝒂𝒑𝒆 𝑿 =

𝒔𝒉𝒂𝒑𝒆 𝑿 𝒊𝒔 𝒂 𝒑𝒐𝒍𝒚𝒈𝒐𝒏

𝒔𝒉𝒂𝒑𝒆 𝑿 𝒉𝒂𝒔 𝟒 𝒔𝒊𝒅𝒆𝒔

𝒔𝒉𝒂𝒑𝒆 𝑿 𝒊𝒔 𝒂𝒏𝒚 𝒇𝒐𝒓𝒎

𝒔𝒉𝒂𝒑𝒆 𝑿 𝒊𝒔 𝒄𝒐𝒏𝒗𝒆𝒙

𝑥

𝑦

𝒔𝒉𝒂𝒑𝒆 𝑿 =

⟹

https://en.wikipedia.org/wiki/Partially_ordered_set

5

2

3

1

4

𝐴

𝐵

𝐶

3

𝐷

𝐴

𝐵

𝐶

3

𝐷

𝑩𝒓𝒆𝒂𝒅 𝑪𝒉𝒆𝒆𝒔𝒆 𝑾𝒊𝒏𝒆

𝒈𝟏 × ×

𝒈𝟐 × ×

𝒈𝟑 × ×

𝒈𝟒 × ∅

{𝐵} {𝐶} {𝑊}

{𝐵, 𝐶} {𝐵,𝑊} {𝐶,𝑊}

{𝐵, 𝐶,𝑊}

⊆

𝑩𝒓𝒆𝒂𝒅 𝑪𝒉𝒆𝒆𝒔𝒆 𝑾𝒊𝒏𝒆

𝒈𝟏 × ×

𝒈𝟐 × ×

𝒈𝟑 × ×

𝒈𝟒 ×

⊆

∅

{𝐵} {𝐶} {𝑊}

{𝐵, 𝐶} {𝐵,𝑊} {𝐶,𝑊}

{𝐵, 𝐶,𝑊}

∅

{𝐵} {𝐶} {𝑊}

{𝐵, 𝐶} {𝐵,𝑊} {𝐶,𝑊}

{𝐵, 𝐶,𝑊}𝑩𝒓𝒆𝒂𝒅 𝑪𝒉𝒆𝒆𝒔𝒆 𝑾𝒊𝒏𝒆

𝒈𝟏 × ×

𝒈𝟐 × ×

𝒈𝟑 × ×

𝒈𝟒 ×

⊆

𝜹(⋅)

𝒈𝟏 {𝐵,𝑊}

𝒈𝟐 {𝐵, 𝐶}

𝒈𝟑 {𝐶,𝑊}

𝒈𝟒 {𝑊}

𝑔

𝛿(𝑔)

∅

{𝐵} {𝐶} {𝑊}

{𝐵, 𝐶} {𝐵,𝑊} {𝐶,𝑊}

{𝐵, 𝐶,𝑊}

⊆

𝜹(⋅)

𝒈𝟏 {𝐵,𝑊}

𝒈𝟐 {𝐵, 𝐶}

𝒈𝟑 {𝐶,𝑊}

𝒈𝟒 {𝑊}

𝒅 = {𝑩}𝒈𝟏 𝒈𝟐

𝑔

𝛿(𝑔)

𝑑

𝑔 𝑑 ⊆ 𝛿(𝑔)

⊑𝜹(⋅)

𝒈𝟏 𝜹 𝒈𝟏𝒈𝟐 𝜹 𝒈𝟐𝒈𝟑 𝜹 𝒈𝟑𝒈𝟒 𝜹 𝒈𝟒

𝑔

𝛿(𝑔)

𝑑

𝑔 𝑑 ⊑ 𝛿(𝑔) (𝒟,⊑)

⊑

(𝓓,⊑)𝜹𝓖

𝐏𝐚𝐭𝐭𝐞𝐫𝐧 𝐬𝐞𝐭𝐮𝐩 ∶(𝓖 , 𝒟,⊑ , 𝜹)𝓖 (𝒟,⊑) 𝜹

𝒅 ∈ 𝓓 𝒈 ∈ 𝓖

𝒅 ⊑ 𝜹 𝒈 𝒅 𝒈 𝒈 𝒅

𝐴

𝐵

𝐶

3

𝐷

𝓖 𝜹(⋅)

𝒈𝟏 { 𝑚1, 𝑚3}

𝒈𝟐 { 𝑚1, 𝑚2}

𝒈𝟑 { 𝑚2, 𝑚3}

𝒈𝟒 {𝑚3}

(𝓖, (℘(ℳ),⊆), 𝜹) 𝑚1 𝑔2𝑚1 ⊆ 𝛿 𝑔2 = 𝑚1, 𝑚2

𝑔3 𝛿 𝑔3 = { 𝑚2, 𝑚3}

∅

{𝑚1} {𝑚2} {𝑚3}

{𝑚1, 𝑚2} {𝑚1, 𝑚3} {𝑚2, 𝑚3}

{𝑚1, 𝑚2, 𝑚3}

ℳ ℘ ℳ ⊆

⊆

𝓖 𝜹(⋅)

𝒈𝟏 𝑎𝑏𝑏

𝒈𝟐 𝑏𝑎𝑎

𝒈𝟑 𝑎

𝒈𝟒 𝑏

𝑏 𝑔2𝑏 ⊑ 𝛿 𝑔2 = 𝑎𝒃𝑏

𝑔3 𝛿 𝑔3 = 𝑎

(𝓖, ( 𝒂, 𝒃 +, ⊑), 𝜹

𝑎𝑎 𝑎𝑏

𝑎 𝑏

𝑏𝑎 𝑏𝑏

𝑏𝑏𝑏𝑎𝑎𝑎𝑏𝑎𝑎 𝑎𝑏𝑏…

… …

𝒂, 𝒃 +

𝒂, 𝒃

⊑

𝓖 𝒙 𝒚

𝒈𝟏 4 4

𝒈𝟐 6 6

𝒈𝟑 8 8

𝒈𝟒 8 2 𝒈𝟒

𝒈𝟏𝒈𝟐

𝒈𝟑

𝓖 𝜹(⋅)

𝒈𝟏 4,4 × [4,4]

𝒈𝟐 6,6 × [6,6]

𝒈𝟑 8,8 × [8,8]

𝒈𝟒 8,8 × [2,2]

(𝓖, (𝑪(ℝ)𝟐, ⊇), 𝜹

𝒄 = 𝟐, 𝟗 × [𝟑, 𝟗]

𝒅 = 𝟑, 𝟕 × [𝟑, 𝟕]

𝒄 ⊇ 𝒅𝒄

𝒅 𝒈𝟏 𝒅 ⊇ 𝜹 𝒈𝟏

𝒈𝟑

𝒈𝟒

𝒈𝟏𝒈𝟐

𝒈𝟑

𝒅𝟑 ≤ 𝒙 ≤ 𝟕𝟑 ≤ 𝒚 ≤ 𝟕

ℝ

𝓖 𝜹(⋅)

𝒈𝟏 ( 4,4 , 0)

𝒈𝟐 ( 6,6 , 0)

𝒈𝟑 ( 8,8 , 0)

𝒈𝟒 ( 8,2 , 0)

(𝓖, (𝒅𝒊𝒔𝒌(ℝ𝟐), ⊇), 𝜹

𝒄 = ( 𝟔, 𝟔 , 𝟑. 𝟓)

𝒅 = ( 𝟓, 𝟓 , 𝟐)

𝒈𝟒

𝒈𝟏

𝒈𝟐

𝒈𝟑

(𝟓, 𝟓) 𝟐𝒅 ≡ { 𝒙, 𝒚 ∈ ℝ∣ 𝒙 − 𝟓 𝟐 + 𝒚 − 𝟓 𝟐 ≤ 𝟒}

ℝ𝟐 𝒄 ⊇ 𝒅𝒄

𝒅 𝒈𝟏 𝒅 ⊇ 𝜹 𝒈𝟏

𝒈𝟑

𝐴

𝐵

𝐶

3

𝐷

𝑑 ∈ 𝒟 𝒢 𝑑

𝑒𝑥𝑡: 𝑑 ↦ {𝑔 ∈ 𝒢 ∣ 𝑑 ⊑ 𝛿(𝑔)}

𝑒𝑥𝑡(𝑑) 𝑑

𝑠𝑢𝑝(𝑑)𝒢 (𝒟,⊑)

𝜹𝑑

𝜹

𝜹

𝑒𝑥𝑡(𝑑)

𝓖 𝜹(⋅)

𝒈𝟏 { 𝑚1, 𝑚3}

𝒈𝟐 { 𝑚1, 𝑚2}

𝒈𝟑 { 𝑚2, 𝑚3}

𝒈𝟒 {𝑚3}

(𝓖, (℘(ℳ),⊆), 𝜹)

𝑒𝑥𝑡 𝑚1 = {𝑔1, 𝑔2} 𝑒𝑥𝑡 𝑏 = {𝑔1, 𝑔2, 𝑔4} 𝑒𝑥𝑡 3,7 × 3,7 = {𝑔1, 𝑔2}

(𝓖, ( 𝒂, 𝒃 +, ⊑), 𝜹

𝓖 𝜹(⋅)

𝒈𝟏 𝑎𝑏𝑏

𝒈𝟐 𝑏𝑎𝑎

𝒈𝟑 𝑎

𝒈𝟒 𝑏

𝓖 𝜹(⋅)

𝒈𝟏 4,4 × [4,4]

𝒈𝟐 6,6 × [6,6]

𝒈𝟑 8,8 × [8,8]

𝒈𝟒 8,8 × [2,2]

(𝓖, (𝑪(ℝ)𝟐, ⊇), 𝜹

𝒈𝟒

𝒈𝟏𝒈𝟐

𝒈𝟑

𝓖 𝜹(⋅)

𝒈𝟏 { 𝑚1, 𝑚3}

𝒈𝟐 { 𝑚1, 𝑚2}

𝒈𝟑 { 𝑚2, 𝑚3}

𝒈𝟒 {𝑚3}

(𝓖, (℘(ℳ),⊆), 𝜹) (𝒟,⊑)

∅

{𝑚2} {𝑚1} {𝑚3}

{𝑚1, 𝑚2} {𝑚2, 𝑚3} {𝑚1, 𝑚3}

{𝑚1, 𝑚2, 𝑚3}

(ℙ𝒆𝒙𝒕, ⊆)

∅

{𝑔1} {𝑔3} {𝑔2}

{𝑔1, 𝑔3, 𝑔4} {𝑔1, 𝑔2} {𝑔2, 𝑔3}

{𝑔1, 𝑔2, 𝑔3, 𝑔4}𝒆𝒙𝒕

ℙ

ℙ𝑒𝑥𝑡: = 𝑒𝑥𝑡 𝐷 = {𝑒𝑥𝑡 𝑑 ∣ 𝑑 ∈ 𝐷}

ℙ

ℙ𝑒𝑥𝑡: = 𝑒𝑥𝑡 𝐷 = {𝑒𝑥𝑡 𝑑 ∣ 𝑑 ∈ 𝐷}

𝓖 𝜹(⋅)

𝒈𝟏 𝑎𝑏𝑏

𝒈𝟐 𝑏𝑎𝑎

𝒈𝟑 𝑎

𝒈𝟒 𝑏

(𝓖, ( 𝒂, 𝒃 +, ⊑), 𝜹) (ℙ𝒆𝒙𝒕, ⊆)

𝒆𝒙𝒕

{𝑔1, 𝑔2, 𝑔4} {𝑔1, 𝑔2, 𝑔3}

{𝑔1} {𝑔2}

∅

𝑎𝑎 𝑎𝑏

𝒂 𝒃

𝑏𝑎 𝑏𝑏

𝑏𝑏𝑏𝑎𝑎𝑎𝑏𝑎𝑎 𝑎𝑏𝑏…

… …

(𝒟,⊑)

𝐴 ⊆ 𝒢 𝒟

𝑐𝑜𝑣: 𝐴 ↦ 𝑑 ∈ 𝒟 ∀𝑔 ∈ 𝐴 𝑑 ⊑ 𝛿 𝑔

𝑐𝑜𝑣(𝐴)

𝛿 𝐴 = {𝛿 𝑔 ∣ 𝑔 ∈ 𝐴} (𝒟,⊑)

𝛿 𝐴 ℓ𝒢 (𝒟,⊑)

𝜹

𝜹

𝑐𝑜𝑣(𝐴)𝐴

𝜹

𝓖 𝜹(⋅)

𝒈𝟏 { 𝑚1, 𝑚3}

𝒈𝟐 { 𝑚1, 𝑚2}

𝒈𝟑 { 𝑚2, 𝑚3}

𝒈𝟒 {𝑚3}

(𝓖, (℘(ℳ),⊆), 𝜹)

𝑐𝑜𝑣 𝑔1, 𝑔2 = 𝑚1 , ∅ 𝑐𝑜𝑣 𝑔1, 𝑔2 =

(𝓖, ( 𝒂, 𝒃 +, ⊑), 𝜹

𝓖 𝜹(⋅)

𝒈𝟏 𝑎𝑏𝑏

𝒈𝟐 𝑏𝑎𝑎

𝒈𝟑 𝑎

𝒈𝟒 𝑏

𝓖 𝜹(⋅)

𝒈𝟏 4,4 × [4,4]

𝒈𝟐 6,6 × [6,6]

𝒈𝟑 8,8 × [8,8]

𝒈𝟒 8,8 × [2,2]

(𝓖, (𝑪(ℝ)𝟐, ⊇), 𝜹

𝒈𝟒

𝒈𝟏𝒈𝟐

𝒈𝟑

𝑎, 𝑏 𝑐𝑜𝑣 𝑔1, 𝑔2 =4,4 (6,6)

∀𝐴, 𝐵 ∈ ℘ 𝒢 𝐴 ⊆ 𝐵 ⇒ 𝑐𝑜𝑣 𝐵 ⊆ 𝑐𝑜𝑣(𝐴)

𝒄𝒐𝒗(℘(𝒟), ⊆)(℘(𝒢), ⊆)

∀𝑐, 𝑑 ∈ 𝐷 𝑐 ⊑ 𝑑 ⇒ 𝑒𝑥𝑡 𝑑 ⊆ 𝑒𝑥𝑡(𝑐)(𝒟,⊑)𝒆𝒙𝒕

(℘(𝒢), ⊆)

𝒄 ∈ 𝒟 𝒅 ∈ 𝒟 ℙ

𝒄 → 𝒅 𝒈 ∈ 𝒢 𝒄 ℙ 𝒅

𝒄 → 𝒅 ⟺ 𝒆𝒙𝒕 𝒄 ⊆ 𝒆𝒙𝒕 𝒅

(𝓖, ( 𝒂, 𝒃 +, ⊑), 𝜹

𝓖 𝜹(⋅)

𝒈𝟏 𝑎𝑏𝑏

𝒈𝟐 𝑏𝑎𝑎

𝒈𝟑 𝑎

𝒈𝟒 𝑏 𝑎𝑎 → 𝑏𝑎

𝑏𝑎𝑎 → 𝑏𝑎

𝑑 ⊑ 𝑐 ⇒ 𝑒𝑥𝑡 𝑐 ⊆ 𝑒𝑥𝑡 𝑑

𝒄 → 𝒅 𝒈 ∈ 𝒢

𝒄 ℙ 𝒅

𝒄𝒐𝒏𝒇 𝒄 → 𝒅 ≔𝒆𝒙𝒕 𝒄 ∩ 𝒆𝒙𝒕 𝒅

𝒆𝒙𝒕 𝒄

(𝓖, ( 𝒂, 𝒃 +, ⊑), 𝜹

𝓖 𝜹(⋅)

𝒈𝟏 𝑎𝑏𝑏

𝒈𝟐 𝑏𝑎𝑎

𝒈𝟑 𝑎

𝒈𝟒 𝑏 𝑎𝑎 → 𝑏𝑎

𝑎 → 𝑏𝒄𝒐𝒏𝒇 𝑎 → 𝑏 66%

𝒄 ∈ 𝒟 𝒅 ∈ 𝒟 ℙ

𝒄 ↔ 𝒅 𝒈 ∈ 𝒢 𝒄 ℙ 𝒅

𝒄 ↔ 𝒅⟺ 𝒄 → 𝒅 𝒄 → 𝒅⟺ 𝒆𝒙𝒕 𝒄 = 𝒆𝒙𝒕 𝒅

𝒄 ⟷ 𝒅 ∀𝒆 ∈ 𝒄, 𝒅 ) 𝒆 ⟷ 𝒅

𝐴

𝒆𝒙𝒕−𝟏 𝑨

𝒎𝒂𝒙(𝒆𝒙𝒕−𝟏 𝑨 )

𝒎𝒊𝒏(𝒆𝒙𝒕−𝟏 𝑨 )

𝒆 ∈ 𝓓 𝒄 ⊑ 𝒆 ⊑ 𝒅

𝒆𝒙𝒕 𝒆 ⊂ 𝑨

𝑨 ⊂ 𝒆𝒙𝒕(𝒆 )

(𝓖, ( 𝒂, 𝒃 +, ⊑), 𝜹

𝓖 𝜹(⋅)

𝒈𝟏 𝑎𝑏𝑏𝑏𝑎

𝒈𝟐 𝑏𝑏𝑏𝑎𝑏𝑏𝑏

𝒈𝟑 𝑎

𝒈𝟒 𝑏

𝑎𝑏𝑏𝑏 𝑏𝑏𝑏𝑎

𝑏𝑏𝑏𝑎𝑏𝑏 𝑏𝑏𝑎

𝑏𝑏𝑎𝑏 𝑏𝑎

𝒆𝒙𝒕{𝑔1, 𝑔2}

𝑎𝑏𝑏𝑏𝑎

𝑏𝑏𝑏𝑎𝑏𝑏𝑏

𝑎 𝑏

𝐴

𝐵

𝐶

3

𝐷

𝐴 ⊆ 𝒢 𝑐𝑜𝑣(𝐴)

𝑐𝑜𝑣∗ 𝐴

𝑐𝑜𝑣∗: 𝐴 ↦ max 𝑐𝑜𝑣 𝐴

𝒄𝒐𝒗 𝑨

𝒄𝒐𝒗∗ 𝑨

𝒄𝒐𝒗 𝑨

𝒄𝒐𝒗∗ 𝑨

𝒄𝒐𝒗 𝑨

𝒄𝒐𝒗∗ 𝑨

(𝓖, (𝒟,⊑), 𝜹)

(𝓖, (𝒟,⊑), 𝜹)

𝐴 ⊆ 𝒢

𝑐𝑜𝑣(𝐴)

𝒄𝒐𝒗 𝑨

𝒊𝒏𝒕(𝑨)

𝐴𝜹

𝜹

𝜹

𝛿 𝐴

𝑖𝑛𝑡𝐴

𝑖𝑛𝑡 𝐴

𝓖 𝜹(⋅)

𝒈𝟏 { 𝑚1, 𝑚3}

𝒈𝟐 { 𝑚1, 𝑚2}

𝒈𝟑 { 𝑚2, 𝑚3}

𝒈𝟒 {𝑚3}

(𝓖, (℘(ℳ),⊆), 𝜹) (𝓖, ( 𝒂, 𝒃 +, ⊑), 𝜹

𝓖 𝜹(⋅)

𝒈𝟏 𝑎𝑏𝑏

𝒈𝟐 𝑏𝑎𝑎

𝒈𝟑 𝑎

𝒈𝟒 𝑏

𝑖𝑛𝑡 ∶ 𝐴 →

𝑔∈𝐴

𝛿(𝑔)

𝑐𝑜𝑣 𝑔1, 𝑔2 = {𝑎, 𝑏}

𝓖 𝒙 𝒚

𝒈𝟏 4 4

𝒈𝟐 6 6

𝒈𝟑 8 8

𝒈𝟒 8 2

𝓖 𝒙 𝒚

𝒈𝟏 4 4

𝒈𝟐 6 6

𝒈𝟑 8 8

𝒈𝟒 8 2

(𝓖, (𝑪(ℝ)𝟐, ⊇), 𝜹 (𝓖, (𝒅𝒊𝒔𝒌(ℝ𝟐), ⊇), 𝜹

𝒈𝟒

𝒈𝟏𝒈𝟐

𝒈𝟑

𝒈𝟒

𝒈𝟏𝒈𝟐

𝒈𝟑

(𝓓,⊑)

(𝓓,⊑)

(𝓖, (𝓓,⊑), 𝜹)

𝓖 𝜹

𝒟

(𝓓,⊑)

(𝓖, (𝒟𝟏, ⊑𝟏), 𝜹𝟏) (𝓖, (𝒟𝟐, ⊑𝟐), 𝜹𝟐)

(𝓖, (𝒟𝟏 × 𝒟𝟐, ⊑× ), 𝜹×)

×

=

𝛿× ≔ (𝛿1, 𝛿2)

•

•

5

2

3

1

4

𝒪 𝛼 ⋅ |𝒢| 𝛼

𝑒𝑥𝑡 ∘ 𝑖𝑛𝑡

𝒪 𝜷 + 𝒢 2 𝜷

𝑒𝑥𝑡 ∘ 𝑖𝑛𝑡

•

ℙ = 𝒢, 𝒟,⊑ , 𝛿

• (𝑒𝑥𝑡, 𝑖𝑛𝑡)

• 𝑒𝑥𝑡 ∘ 𝑖𝑛𝑡

• 𝑖𝑛𝑡 ∘ 𝑒𝑥𝑡

•

•

𝑑 ∈ 𝒟

ℙ 𝑋 = 𝑑 =𝜙 𝑑

𝑐∈𝒟 𝜙(𝑐)

𝜙

d ↦ |𝑒𝑥𝑡(𝑑)|𝒟,⊑

𝑔~↓𝛿 𝑔

𝑒∈𝒢 ↓𝛿 𝑒

𝑑 ∈↓ 𝛿 𝑔

𝑔

5

2

3

1

4

⟹

(𝓓,⊑)

𝜹

𝓖

𝜹

𝜹

ℙ = (𝓖, (𝓓,⊑), 𝜹)

𝒅 ∈ 𝓓

𝒈 ∈ 𝓖 𝒅 ⊑ 𝜹 𝒈

https://belfodilaimene.github.io/pattern-setups-tutorial/