补集bǔjí

Khai niem co ban

Phan bu cua tap hop A, ky hieu la $\complement_U A$ hoac $\overline{A}$ hoac $A^c$, la tap hop chua tat ca cac phan tu trong tap hop vu tru U ma KHONG thuoc A.

Dinh nghia toan hoc

$\complement_U A = \{x | x \in U \text{ va } x \notin A\}$

Phan bu chua chinh xac nhung phan tu thuoc tap hop vu tru nhung khong thuoc A.

Cac dang ky hieu

• $\complement_U A$ - Ky hieu chuan nhan manh tap hop vu tru
• $\overline{A}$ - Ky hieu gach ngang tren
• $A^c$ hoac $A'$ - Ky hieu chi so tren
• $U - A$ - Ky hieu hieu cua tap hop

Bieu dien truc quan

Trong bieu do Venn, phan bu la vung ben ngoai tap hop A nhung ben trong tap hop vu tru.

  U: [#############]
     [####]  A  [  ]

Vung to mau [####] bieu dien $\complement_U A$.

Tinh chat quan trong

1. Phan bu cua phan bu

$\complement_U(\complement_U A) = A$

2. Phan bu cua tap hop vu tru

$\complement_U U = \emptyset$

3. Phan bu cua tap rong

$\complement_U \emptyset = U$

4. Hop voi phan bu

$A \cup \complement_U A = U$

5. Giao voi phan bu

$A \cap \complement_U A = \emptyset$

6. Dinh luat De Morgan

$\complement_U(A \cup B) = \complement_U A \cap \complement_U B$ $\complement_U(A \cap B) = \complement_U A \cup \complement_U B$

Vi du

Vi du 1: Tap hop huu han

Cho: U = {1, 2, 3, 4, 5, 6}, A = {2, 4, 6}

Tim: $\complement_U A$

Giai: Phan tu trong U nhung khong trong A: 1, 3, 5

Dap an: $\complement_U A$ = {1, 3, 5}

Vi du 2: Tap hop so thuc

Cho: U = ℝ, A = {x | x ≥ 2}

Tim: $\complement_U A$

Giai: So thuc KHONG ≥ 2, nghia la < 2

Dap an: $\complement_U A$ = {x | x < 2} = (-∞, 2)

Vi du 3: Phan bu cua khoang

Cho: U = ℝ, A = (-1, 3]

Tim: $\complement_U A$

Giai: Tat ca so thuc ngoai (-1, 3]

Dap an: $\complement_U A$ = (-∞, -1] ∪ (3, +∞)

Bai tap CSCA

💡 Luu y: Cac bai tap sau duoc thiet ke dua tren de cuong thi CSCA.

Vi du 1: Co ban (Do kho ★★☆☆☆)

Neu U = {1, 2, 3, 4, 5} va A = {1, 3, 5}, tim $\complement_U A$.

Lua chon:

• A. {1, 3, 5}
• B. {2, 4}
• C. {1, 2, 3, 4, 5}
• D. ∅

Giai: Phan tu trong U nhung khong trong A: 2, 4

Dap an: B

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Vi du 2: Trung binh (Do kho ★★★☆☆)

Cho U = ℝ, A = {x | x² - 4 ≤ 0}, tim $\complement_U A$.

Giai:

Truoc tien, giai bat phuong trinh: $x² - 4 ≤ 0$ $(x-2)(x+2) ≤ 0$ $A = [-2, 2]$

Phan bu la tat ca so thuc ngoai khoang nay: $\complement_U A = (-\infty, -2) \cup (2, +\infty)$

Dap an: $(-\infty, -2) \cup (2, +\infty)$

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Vi du 3: Nang cao (Do kho ★★★★☆)

Neu U = ℝ, A = {x | x > 1}, B = {x | x > 2}, tim $\complement_U A \cup B$.

Giai:

$\complement_U A$ = {x | x ≤ 1} = (-∞, 1] B = {x | x > 2} = (2, +∞)

$\complement_U A \cup B = (-\infty, 1] \cup (2, +\infty)$

Dap an: $(-\infty, 1] \cup (2, +\infty)$

Dinh luat De Morgan chi tiet

Dinh luat 1: Phan bu cua hop

$\complement_U(A \cup B) = \complement_U A \cap \complement_U B$

Vi du: Neu A = {1, 2}, B = {2, 3}, U = {1, 2, 3, 4}

• A ∪ B = {1, 2, 3}
• $\complement_U(A \cup B)$ = {4}
• $\complement_U A$ = {3, 4}, $\complement_U B$ = {1, 4}
• $\complement_U A \cap \complement_U B$ = {4} ✓

Dinh luat 2: Phan bu cua giao

$\complement_U(A \cap B) = \complement_U A \cup \complement_U B$

Loi thuong gap

❌ Loi 1: Quen tap hop vu tru

Sai: $\complement A$ = {tat ca phan tu khong trong A} ✗

Dung: $\complement_U A$ = {phan tu trong U nhung khong trong A} ✓

❌ Loi 2: Sai bien khoang

Sai: Neu A = [1, 3], thi $\complement_\mathbb{R} A$ = (-∞, 1] ∪ [3, +∞) ✗

Dung: $\complement_\mathbb{R} A$ = (-∞, 1) ∪ (3, +∞) ✓

❌ Loi 3: Sai dau trong De Morgan

Sai: $\complement(A \cup B)$ = $\complement A \cup \complement B$ ✗

Dung: $\complement(A \cup B)$ = $\complement A \cap \complement B$ ✓

Meo hoc tap

1. ✅ Luon xac dinh U truoc: Tap hop vu tru quyet dinh phan bu
2. ✅ Dao bien cho khoang: Mo ↔ dong khi lay phan bu
3. ✅ Nam vung dinh luat De Morgan: "Pha gach ngang, doi dau"
4. ✅ Phan bu kep tra ve ban dau: $\complement(\complement A) = A$

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💡 Meo thi: Khi lay phan bu cua khoang, nho rang: bien dong thanh mo, va mo thanh dong!