more math proofs | Chelsea Zou

Groups (G, *)

A group is a set \( G \) together with operation \( * \colon G \times G \to G \) (takes two elements and returns one) satisfying the following axioms:

1. Associativity:
\( (a * b) * c = a * (b * c) \quad \quad \forall a,b,c \in G\)

2. Identity:
\(\exists e \) such that \(a * e = a \quad \quad \forall a,e \in G) \)

3. Inverse:
\(\exists a^{-1}\) such that \( a * a^{-1} = e \quad \quad \forall a, a^{-1}, e \in G \)

**4. Commutative (Abelian Group):
\(a * b = b * a \quad \quad \forall a,b \in G\)

Fields (F, +, \( \cdot \))

A field is a set \( F \) with two operations
\( + \colon F \times F \to F \)
\( \cdot \colon F \times F \to F \)

1. \((F, +)\) is an abelian group:
\( a + b = b + a \), associativity, identity \(0\), and additive inverse \( -a \quad \forall a, b \in F \)

2. \((F \setminus \{0\}, \cdot)\) is an abelian group:
\( a \cdot b = b \cdot a \), associativity, identity \(1\), and multiplicative inverse \( a^{-1} \quad \forall a \neq 0 \)

3. Distributivity:
\( a \cdot (b + c) = a \cdot b + a \cdot c \quad \quad \forall a, b, c \in F \)

Vector Spaces (V)

A vector space over a field \( F \) is a set \( V \) equipped with:

• Closed under Addition \( + \colon V \times V \to V \), where \( u + v \in V \) for all \( u, v \in V \).
• Closed under Scalar multiplication \( \cdot \colon F \times V \to V \), where \( \lambda v \in V \) for each \( \lambda \in F \), \( v \in V \).

1. Commutativity:
\( u + v = v + u \quad \forall u, v \in V \)

2. Associativity:
\( (u + v) + w = u + (v + w) \quad \forall u, v, w \in V \)
\( (ab)v = a(bv) \quad \forall a, b \in F, \, v \in V \)

3. Additive Identity:
\( \exists 0 \in V \) such that \( v + 0 = v \quad \forall v \in V \)

4. Additive Inverse:
\( \forall v \in V, \, \exists w \in V \) such that \( v + w = 0 \)

5. Multiplicative Identity:
\( 1v = v \quad \forall v \in V \)

6. Distributivity:
\( a(u + v) = au + av \quad \forall a \in F, \, u, v \in V \)
\( (a + b)v = av + bv \quad \forall a, b \in F, \, v \in V \)

Modular Arithmetic

The set \( \mathbb{Z}/n\mathbb{Z} = \{0, 1, 2, \ldots , n-1\} \) where each element corresponds to the set of integers that when divided by n, has that remainder.

For n = 3, we have \( \mathbb{Z}/3\mathbb{Z} = \{[0], [1], [2]\} \)
• \([0] = \{0,3,6,9, \ldots\}\)
• \([1] = \{1,4,7,10, \ldots\}\)
• \([2] = \{2,5,8,11, \ldots\}\)

Field Property:
If \( p \) is prime, then \( \mathbb{Z}/p\mathbb{Z} \) forms a field \( \mathbb{F} \): every nonzero element has a multiplicative inverse.

Injectivity:
If f(x) = f(y), then x = y