Set Identities formulas

1.

$A \sub I$

Sets

2.

$A \sub A$

Sets

3.

Empty set

$\emptyset \sub A$

Sets

4.

Union of sets

$A \cup B= \{x : x \in A \; \text{or} \; x \in B \}$

Sets

5.

Intersection of sets

$A \cap B= \{x : x \in A \; \text{and} \; x \in B \}$

Sets

6.

Difference of sets

$A \setminus B = \{ x : x\in A \, \text{and} \, x\notin B\}$

Sets

7.

Set commutativity

$A \cup B = B \cup A$

Sets

8.

Set commutativity

$A \cap B = B \cap A$

Sets

9.

Set associativity

$A \cup \left(B \cup C \right) = \left( A \cup B \right) \cup C$

Sets

10.

Set associativity

$A \cap \left(B \cap C \right) = \left( A \cap B \right) \cap C$

Sets

11.

Set distributivity

$A \cup \left(B \cap C\right) = \left(A \cup B \right) \cap \left(A \cup C \right)$

Sets

12.

Set distributivity

$A \cap \left(B \cup C\right) = \left(A \cap B \right) \cup \left(A \cap C \right)$

Sets

13.

Idempotency

$A \cup A = A$

Sets

14.

Idempotency

$A \cap A = A$

Sets

15.

Domination

$A \cap \empty = A$

Sets

16.

Domination

$A \cup I = I$

Sets

17.

identity

$A \cup \empty = A$

Sets

18.

Identity

$A \cap I = A$

Sets

19.

Complement

$A^\prime=\{x \in I : x \notin A\}$

Sets

20.

Complement of union

$A \cup A^\prime = I$

Sets

21.

Complement of intersection

$A \cap A^\prime=\empty$

Sets

22.

De morgan's law

$\left(A \cup B\right)^\prime=A^\prime \cap B^\prime$

Sets

23.

De morgan's law

$\left(A \cap B\right)^\prime=A^\prime \cup B^\prime$

Sets

24.

$B \setminus A = B \setminus \left(A \cap B\right)$

Sets

25.

$B \setminus A^\prime = B \cap A^\prime$

Sets

26.

$A \setminus A = \empty$

Sets

27.

$\left(A \setminus B\right) \cap C = \left(A \cap B \right) \setminus \left(B \cap C \right)$

Sets