Randhiredu

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Which of the following sets is bounded below but not
bounded above?
1. N
2. Z
3.Q
4.R

Definition of a linear space. A linear space or a vector space over a field of scalars
(usually, the field of real numbers or the field of complex numbers) is a set V of elements x, y, z, ... (also called vectors) of any nature for which the following conditions hold:
I. There is a rule that establishes correspondence between any pair of elements x, y ∈ V and a third element z ∈ V, called the sum of the elements x, y and denoted by z = x + y.
II. There is a rule that establishes correspondence between any pair x, λ, where x is an
element of V and λ is a scalar, and an element u ∈ V, called the product of a scalar λ
and a vector x and denoted by u = λx.
III. The following eight axioms are assumed for the above two operations:
1. Commutativity of the sum: x + y = y + x.
2. Associativity of the sum: (x + y) + z = x + (y + z).
3. There is a zero element 0 such that x + 0 = x for any x.
4. For any element x there is an opposite element x′ such that x + x′ = 0.
5. A special role of the unit scalar 1: 1 ⋅ x = x for any element x.
6. Associativity of the multiplication by scalars: λ(µx) = (λµ)x.
7. Distributivity with respect to the addition of scalars: (λ + µ)x = λx + µx.
8. Distributivity with respect to a sum of vectors: λ(x + y) = λx + λy

Main properties of operations with vectors.
1. a + b = b + a (commutativity).
2. a + (b + c) = (a + b) + c (associativity of addition).
3. a + 0 = a (existence of zero vector).
4. a + (–a) = 0 (existence of opposite vector).
5. λ(a + b) = λa + λb (distributivity with respect to addition of vectors).
6. (λ + µ)a = λa + µa (distributivity with respect to addition of constants).
7. λ(µa) = (λµ)a (associativity of product).
8. 1a = a (multiplication by unity).