Ensembles

Définition

\[ \begin{aligned} Les \; & \blue{\mathbb{N}}aturels \\ Les \; & \blue{\mathbb{Z}}'entiers \\ Les \; & \blue{\mathbb{Q}}uotiens \\ Les \; & \blue{\mathbb{R}}éels \\ Les \; & \blue{\mathbb{C}}omplexes \end{aligned} \]
\[ \begin{aligned} Les \; \mathbb{Z}'entiers \, \orange{\mathbb{N}}on\, \orange{\mathbb{N}}égatifs \end{aligned} \]
\[ A \in B \] A est dans B
\[ A \subseteq B \] A est faisable avec B
\[ \varnothing \subseteq \varnothing \\[2em] \varnothing \notin \{ 1 \} \\[2em] \varnothing \subseteq \{ 1 \} \]

Exemple

Ensemble

\[ \begin{aligned} & A = \{a, b, c\} \\[0.5em] & |A| = 3 \end{aligned} \]
a b c

Produit cartésien

\[ \begin{aligned} & A \times A = A^2 = \{(a, a), (a, b), ..., (c, c)\} \\[0.5em] & |A|^2 = 9 \\ \end{aligned} \]
a b c
a (a,a) (a,b) (a,c)
b (b,a) (b,b) (b,c)
c (c,a) (c,b) (c,c)

Parties

\[ \begin{aligned} & \mathscr{P}(A) = \{\varnothing, \{a\}, ..., \{a, b, c\}\} \\[1em] & | \mathscr{P}(A) | = 2^{|A|} = 2^3 = 8 \end{aligned} \]
a b c
∅
✔ {c}
✔ {b}
✔ ✔ {b, c}
✔ {a}
✔ ✔ {a, c}
✔ ✔ {a, b}
✔ ✔ ✔ {a, b, c}

Appartenances

\[ \begin{aligned} & A = \{a, b, c\} \quad; & B = \{a, b\} \quad; && C = \{\{a\}, \{b\}\} \end{aligned} \]\[ \begin{aligned} & B : \quad & B \subseteq A \quad; && B \nsubseteq \mathscr{P}(A) \quad; && B \in \mathscr{P}(A) \quad; && B \notin A \\[0.5em] & C : \quad & C \nsubseteq A \quad; && C \subset \mathscr{P}(A) \quad; && C \notin\mathscr{P}(A) \quad; && C \notin A \\ \end{aligned} \]

Opérateurs logiques

Set Set Math Logic Prog Bool door table
∩ ∧ AND && • 00 0
01 0
10 0
11 1
∪ ∨ OR || + 00 0
01 1
10 1
11 1
Δ ⊖ XOR ⊕ 00 0
01 1
10 1
11 0
XNOR ⊙ 00 1
01 0
10 0
11 1

Karnaugh

A
0 1
B 2 3

A
0 2 3 1
C 4 6 7 5
B

A
0 4 12 8
1 5 13 9 D
C 3 7 15 11
2 6 14 10
B