_laws | math-debrief

Laws

$A$ 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.

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

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)

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

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

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: $\frac{a}{b} = a \frac{1}{b}$

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)

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

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