Let's solve the problem step by step given the values [tex]\( a = -13 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -11 \)[/tex].
### (i) Verification of [tex]\( a + b = b + a \)[/tex]
First, we need to verify the commutative property of addition, which states that [tex]\( a + b \)[/tex] should be equal to [tex]\( b + a \)[/tex].
1. Calculate [tex]\( a + b \)[/tex]:
[tex]\[
a + b = -13 + 5 = -8
\][/tex]
2. Calculate [tex]\( b + a \)[/tex]:
[tex]\[
b + a = 5 + -13 = -8
\][/tex]
Now, compare the two results:
[tex]\[
a + b = -8 \quad \text{and} \quad b + a = -8
\][/tex]
Since [tex]\(-8 = -8\)[/tex], we can confirm that [tex]\( a + b = b + a \)[/tex].
### (ii) Verification of [tex]\( (a + b) + c = a + (b + c) \)[/tex]
Next, we need to verify the associative property of addition, which states that [tex]\( (a + b) + c \)[/tex] should be equal to [tex]\( a + (b + c) \)[/tex].
1. Calculate [tex]\( (a + b) + c \)[/tex]:
- First, find [tex]\( a + b \)[/tex]:
[tex]\[
a + b = -13 + 5 = -8
\][/tex]
- Next, add the result to [tex]\( c \)[/tex]:
[tex]\[
(a + b) + c = -8 + (-11) = -19
\][/tex]
2. Calculate [tex]\( a + (b + c) \)[/tex]:
- First, find [tex]\( b + c \)[/tex]:
[tex]\[
b + c = 5 + (-11) = -6
\][/tex]
- Next, add the result to [tex]\( a \)[/tex]:
[tex]\[
a + (b + c) = -13 + (-6) = -19
\][/tex]
Now, compare the two results:
[tex]\[
(a + b) + c = -19 \quad \text{and} \quad a + (b + c) = -19
\][/tex]
Since [tex]\(-19 = -19\)[/tex], we can confirm that [tex]\( (a + b) + c = a + (b + c) \)[/tex].
### Conclusion
Based on the calculations above, we have verified that both properties hold true:
1. [tex]\( a + b = b + a \)[/tex]
2. [tex]\( (a + b) + c = a + (b + c) \)[/tex]
Therefore, the given values [tex]\( a = -13 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -11 \)[/tex] satisfy the commutative and associative properties of addition.
