Question

give me a solution fo 3.2, 3.3 definition 3.1 A ‘ring’ (R, +, ·) consists of a set R along with two binary operations + and · (called ‘addition’ and ‘multiplication’), satisfying the following properties. (i) Associativity for addition: ∀a, b, c ∈ R (a + b) + c = a + (b + c) (ii) Existence of additive identity: ∃0R ∈ R ∀a ∈ R a + 0R = 0R + a = a (iii) Existence of additive inverses: ∀a ∈ R ∃z ∈ R a + z = z + a = 0R (iv) Commutativity for addition: ∀a, b ∈ R a + b = b + a (v) Associativity for multiplication: ∀a, b, c ∈ R (a · b) · c = a · (b · c) (vi) Existence of multiplicative identity: ∃1R ∈ R ∀a ∈ R a · 1R = 1R · a = a (vii) Distributivity: ∀a, b, c ∈ R a · (b + c) = a · b + a · c (a + b) · c = a · c + b · c

Answer 1

A 'ring' (R, +, ·) is a set R along with two binary operations + and · that satisfy specific properties including associativity, existence of identities, existence of inverses, commutativity, and distributivity.

Step-by-step explanation:

A 'ring' (R, +, ·) consists of a set R along with two binary operations + and · (called 'addition' and 'multiplication'), satisfying the following properties:

1. Associativity for addition: ∀a, b, c ∈ R (a + b) + c = a + (b + c)
2. Existence of additive identity: ∃0R ∈ R ∀a ∈ R a + 0R = 0R + a = a
3. Existence of additive inverses: ∀a ∈ R ∃z ∈ R a + z = z + a = 0R
4. Commutativity for addition: ∀a, b ∈ R a + b = b + a
5. Associativity for multiplication: ∀a, b, c ∈ R (a · b) · c = a · (b · c)
6. Existence of multiplicative identity: ∃1R ∈ R ∀a ∈ R a · 1R = 1R · a = a
7. Distributivity: ∀a, b, c ∈ R a · (b + c) = a · b + a · c (a + b) · c = a · c + b · c