Sets: intersection ∩ and union ∪ in 2 minutes

Intersection: $A \cap B$

$A \cap B = \{x \, | \, x \in A \text{ AND } x \in B\}$

Elements in $A$ AND in $B$ at the same time.

Example: $A = \{1, 2, 3, 4\}$, $B = \{3, 4, 5, 6\}$ ⇒ $A \cap B = \{3, 4\}$.

Union: $A \cup B$

$A \cup B = \{x \, | \, x \in A \text{ OR } x \in B\}$

Elements in $A$ OR in $B$ (or both).

Example: $A \cup B = \{1, 2, 3, 4, 5, 6\}$.

On intervals

$[1, 5] \cap [3, 7] = [3, 5]$

$[1, 5] \cup [3, 7] = [1, 7]$

Special cases

• $A \cap A = A$, $A \cup A = A$.
• $A \cap \emptyset = \emptyset$, $A \cup \emptyset = A$.
• If $A \subset B$: $A \cap B = A$, $A \cup B = B$.

Complement

$\overline{A} = \{x \, | \, x \notin A\}$

In a reference set $E$, $\overline{A} = E \setminus A$.

De Morgan’s laws

$\overline{A \cap B} = \overline{A} \cup \overline{B}$

$\overline{A \cup B} = \overline{A} \cap \overline{B}$

“Negation swaps AND and OR”.

Venn diagram

Two overlapping circles:

• Intersection = common zone (middle).
• Union = whole picture.

Common mistakes

1. Confusing $\cap$ (and) with $\cup$ (or).
2. Writing $A \cap B = A \cup B$ carelessly.
3. Using $\in$ (element) and $\subset$ (subset) interchangeably.

?À toi de jouer

A = [0, 4] and B = [2, 6]. A ∩ B = ?

A.[2, 4]B.[0, 6]C.[0, 4]D.[2, 6]

Valider