# \_laws

## Laws

[https://web2.0calc.com/formulary/math/algebra/basic-identities](https://web2.0calc.com/formulary/math/algebra/basic-identities)

## Axioms

$$A$$ and $$B$$

$$
\overline{A\cup B} \\
\varnothing = \{x:x\not\in X\}
$$

Complement: $$\overline{A\cup B}$$

A∪(B∪C)=(A∪B)∪C, A∩(B∩C)=(A∩B)∩C A∪(B∩C)=(A∪B)∩(A∪C), A∩(B∪C)=(A∩B)∪(A∩C)

A={a,b,c,d} B={c,d,e,f} A ∪ B = {a,b,c,d,e,f} A ∩ B = {c,d} A - B = {a,b} ≠ B - A = {e,f} C(A)={}

op = ∪, ∩ set operartion

* **Commutativity**: $$A\cup B \equiv B\cup A$$
* **Associativity**, e.g. $$(A\cup B)\cup C \equiv A\cup (B\cup C)$$
* **Distributivity**, e.g. $$A \cup (B \cap C) \equiv (A \cup B) \cap (A\cup C)$$
* **Idempotency**: $$\forall A: A\cup A \equiv A, A\cap A \equiv A$$
* **Identity**: $$\forall A:\ A\cup \varnothing = A, A\cap \varnothing = \varnothing, A\cup \mathcal{U}=\mathcal{U}, A\cap \mathcal{U}=A$$
* **Transitivity**: $$(A \le B \le C) \to (A \le C)$$
* **Involution**: $$\forall A:A \equiv \lnot \lnot A$$
* **De Morgan's Law**: supports in proving tautologies and contradiction.

### Maths › Algebra › Basic Identities

#### Basic Identities

$$\forall a,b,c,x \in \mathbb{R}$$

* Additive Identity: $$a + \epsilon = a$$
* Additive Inverse: $$a + (-a) = \epsilon = 0$$
* Associativity: $$(a + b) + c = a + (b + c)$$
* Commutativity: $$a + b = b + a$$
* Subtraction: $$a - b = a + (-b)$$
* Multiplicative Identity: $$a * 1 = a$$
* Multiplicative Inverse: $$a * (1/a) = 1 (if a is not 0)$$
* Multiplication times $$0: a * 0 = 0$$
* Associative of Multiplication: $$(a * b) * c = a * (b * c)$$
* Commutative of Multiplication: $$a * b = b * a$$
* Distributivity: $$a(b + c) = ab + ac$$
* Division: $$\frac{a}{b} = a \frac{1}{b}$$

#### Exponents

$$
x^a * x^b = x^{a+b}   \\
x^a / x^b = x^{a-b}   \\
a^x \ b^x = {ab}^x   \\
a^x / b^x = {(a/b)}^x   \\
({x^a})^b = x^{ab} = {(x^b)}^a \\
2^{k} = 2^{k-1} + 2^{k-1} = 2 \cdot 2^{k-1} \\
x^{a / b} = \sqrt[b] {x}^{a} \\
\sqrt [3] {x}^{5} \\
\ \\
Logarithms \\
\ \\
Definition: x = logb( a ) \iff b^x = a    (if a,b>0 and b ≠ 1) \\
\\ \
Laws: \\
log(a * b) = log(a) + log(b) \\
log( a^b ) = b * log(a) \\
log( a/b ) = log(a) - log(b)
$$

#### Identities

* $$\Huge x^{\frac{a}{b}} = \sqrt[a] {x}^{b}$$

$$
\Huge
x^{a/b} = \sqrt[a]x^b
$$
