Determine whether the set equipped with the given operations is a vector space. For those that

Answer:
- In Option 1, $V$ is not a vector space, and Axioms 7, 8, and 9 fail to hold.
- In Option 2, Axiom 6 fails to hold, but the specific violation is not specified.
- In Option 3, $V$ is a vector space.
- In Option 4, $V$ is not a vector space, and Axiom 10 fails to hold.
- In Option 5, Axioms 6-10 fail to hold, but the specific violations are not provided.
Explanation:
To determine if the set of all real numbers, denoted as $ℝ$, with addition and multiplication operations forms a vector space, we need to check if all the vector space axioms hold. Let's examine each option and identify which axioms fail in the cases where the set is not a vector space.
Option 1: $V$ is not a vector space, and Axioms 7, 8, and 9 fail to hold.
The vector space axioms are as follows:
Axiom 1: Closure under addition.
Axiom 2: Associativity of addition.
Axiom 3: Existence of additive identity.
Axiom 4: Existence of additive inverse.
Axiom 5: Closure under scalar multiplication.
Axiom 6: Associativity of scalar multiplication.
Axiom 7: Distributivity of scalar sums.
Axiom 8: Distributivity of vector sums.
Axiom 9: Compatibility of scalar multiplication with field multiplication.
Axiom 10: Identity element of scalar multiplication.
To verify Axioms 7, 8, and 9, we need to check if the specified operations satisfy the given properties. Let's go through them:
Axiom 7: Distributivity of scalar sums
$\forall a,b\in ℝ,\forall x\in V:(a+b)x=ax+bx$
Consider $a=2,b=3,x=4$. Then we have:
$(2+3)·4=5·4=20$
$2·4+3·4=8+12=20$
The result holds, so Axiom 7 is satisfied.
Axiom 8: Distributivity of vector sums
$\forall a\in ℝ,\forall x,y\in V:a(x+y)=ax+ay$
Consider $a=2,x=3,y=4$. Then we have:
$2·(3+4)=2·7=14$
$2·3+2·4=6+8=14$
The result holds, so Axiom 8 is satisfied.
Axiom 9: Compatibility of scalar multiplication with field multiplication
$\forall a,b\in ℝ,\forall x\in V:(ab)x=a(bx)$
Consider $a=2,b=3,x=4$. Then we have:
$(2·3)·4=6·4=24$
$2·(3·4)=2·12=24$
The result holds, so Axiom 9 is satisfied.
Therefore, in Option 1, the set $V$ is not a vector space, and Axioms 7, 8, and 9 fail to hold.
Let's continue analyzing the remaining options.
Option 2: $V$ is not a vector space, and Axiom 6 fails to hold.
In this case, Axiom 6, which states associativity of scalar multiplication, is violated. However, it is not specified how Axiom 6 fails. Since the information provided is insufficient, we cannot determine the specific condition under which Axiom 6 is violated.
Option 3: $V$ is a vector space.
In this option, it is stated that $<br>V$ is a vector space. Therefore, all the vector space axioms hold for the set of real numbers with the specified operations.
Option 4: $V$ is not a vector space, and Axiom 10 fails to hold.
Axiom 10, which refers to the identity element of scalar multiplication, is violated in this case. However, it is not specified how Axiom 10 fails. Without further information, we cannot determine the exact condition that causes Axiom 10 to be violated.
Option 5: $V$ is not a vector space, and Axioms 6-10 fail to hold.
In this option, it is stated that Axioms 6-10 fail to hold. Similar to Option 2, the specific details of which axioms fail and under what conditions are not provided. Thus, we cannot determine the exact nature of the violations.

The vector space axioms are as follows:
1. Closure under addition: $\forall 𝐮,𝐯\in V,𝐮+𝐯\in V$.
2. Associativity of addition: $\forall 𝐮,𝐯,𝐰\in V,(𝐮+𝐯)+𝐰=𝐮+(𝐯+𝐰)$.
3. Commutativity of addition: $\forall 𝐮,𝐯\in V,𝐮+𝐯=𝐯+𝐮$.
4. Identity element of addition: $\exists \mathbf{0}\in V$ such that $\forall 𝐮\in V,𝐮+\mathbf{0}=𝐮$.
5. Inverse elements of addition: $\forall 𝐮\in V$, there exists $-𝐮\in V$ such that $𝐮+(-𝐮)=\mathbf{0}$.
6. Closure under scalar multiplication: $\forall 𝐮\in V$ and $\forall c\in ℝ$, $c𝐮\in V$.
7. Associativity of scalar multiplication: $\forall 𝐮\in V$ and $\forall c,d\in ℝ$, $(cd)𝐮=c(d𝐮)$.
8. Distributivity of scalar sums: $\forall 𝐮\in V$ and $\forall c,d\in ℝ$, $(c+d)𝐮=c𝐮+d𝐮$.
9. Distributivity of vector sums: $\forall 𝐮,𝐯\in V$ and $\forall c\in ℝ$, $c(𝐮+𝐯)=c𝐮+c𝐯$.
10. Identity element of scalar multiplication: $\forall 𝐮\in V$, $1𝐮=𝐮$.
Now let's evaluate each given statement and identify the vector space axioms that are false:
1. $V$ is not a vector space, and Axioms 7, 8, and 9 fail to hold.
2. $V$ is not a vector space, and Axiom 6 fails to hold.
3. $V$ is a vector space.
4. $V$ is not a vector space, and Axiom 10 fails to hold.
5. $V$ is not a vector space, and Axioms 6-10 fail to hold.
Therefore, the correct answer is statement 3: $V$ is a vector space.