Answer :
Let's analyze each statement given for the complex numbers [tex]\( x = a + b i \)[/tex], [tex]\( y = c + d i \)[/tex], and [tex]\( z = f + g i \)[/tex] to determine their truth values.
1. Commutativity of Addition:
[tex]\( x + y = y + x \)[/tex]
Addition of complex numbers is commutative:
[tex]\[ x + y = (a + bi) + (c + di) = (a + c) + (b + d)i \][/tex]
[tex]\[ y + x = (c + di) + (a + bi) = (c + a) + (d + b)i \][/tex]
Since addition in the real and imaginary parts is commutative, [tex]\( x + y = y + x \)[/tex] holds true.
This statement is true.
2. Associativity of Multiplication:
[tex]\( (x \times y) \times z = x \times (y \times z) \)[/tex]
Multiplication of complex numbers is associative:
[tex]\[ (x \times y) = (a + bi)(c + di) \][/tex]
By expanding, we get:
[tex]\[ (ac - bd) + (ad + bc)i \][/tex]
Similarly, we compute [tex]\( y \times z \)[/tex] and then multiply by [tex]\( x \)[/tex].
Without loss of generality, because each multiplication step follows associative properties, this leads us to the fact that:
[tex]\[ (x \times y) \times z = x \times (y \times z) \][/tex]
This statement is true.
3. Commutativity of Subtraction:
[tex]\( x - y = y - x \)[/tex]
Subtraction of complex numbers is not commutative:
[tex]\[ x - y = (a + bi) - (c + di) = (a - c) + (b - d)i \][/tex]
[tex]\[ y - x = (c + di) - (a + bi) = (c - a) + (d - b)i \][/tex]
Since [tex]\( a - c \neq c - a \)[/tex] and [tex]\( b - d \neq d - b \)[/tex], [tex]\( x - y \neq y - x \)[/tex].
This statement is false.
4. Associativity of Addition:
[tex]\( (x + y) + z = x + (y + z) \)[/tex]
Addition of complex numbers is associative:
[tex]\[ (x + y) + z = ((a + bi) + (c + di)) + (f + gi) = (a + c + f) + (b + d + g)i \][/tex]
[tex]\[ x + (y + z) = (a + bi) + ((c + di) + (f + gi)) = (a + c + f) + (b + d + g)i \][/tex]
Since both sides equate to [tex]\( (a + c + f) + (b + d + g)i \)[/tex], the property holds.
This statement is true.
5. Associativity of Subtraction:
[tex]\( (x - y) - z = x - (y - z) \)[/tex]
Subtraction of complex numbers is not associative:
[tex]\[ (x - y) - z = ((a + bi) - (c + di)) - (f + gi) = (a - c - f) + (b - d - g)i \][/tex]
[tex]\[ x - (y - z) = (a + bi) - ((c + di) - (f + gi)) = (a - c + f) + (b - d + g)i \][/tex]
Therefore, [tex]\( (a - c - f) + (b - d - g)i \)[/tex] does not equal [tex]\( (a - c + f) + (b - d + g)i \)[/tex].
This statement is false.
Based on the analysis:
- True: [tex]\( x + y = y + x \)[/tex], [tex]\( (x \times y) \times z = x \times (y \times z) \)[/tex], [tex]\( (x + y) + z = x + (y + z) \)[/tex]
- False: [tex]\( x - y = y - x \)[/tex], [tex]\( (x - y) - z = x - (y - z) \)[/tex]
1. Commutativity of Addition:
[tex]\( x + y = y + x \)[/tex]
Addition of complex numbers is commutative:
[tex]\[ x + y = (a + bi) + (c + di) = (a + c) + (b + d)i \][/tex]
[tex]\[ y + x = (c + di) + (a + bi) = (c + a) + (d + b)i \][/tex]
Since addition in the real and imaginary parts is commutative, [tex]\( x + y = y + x \)[/tex] holds true.
This statement is true.
2. Associativity of Multiplication:
[tex]\( (x \times y) \times z = x \times (y \times z) \)[/tex]
Multiplication of complex numbers is associative:
[tex]\[ (x \times y) = (a + bi)(c + di) \][/tex]
By expanding, we get:
[tex]\[ (ac - bd) + (ad + bc)i \][/tex]
Similarly, we compute [tex]\( y \times z \)[/tex] and then multiply by [tex]\( x \)[/tex].
Without loss of generality, because each multiplication step follows associative properties, this leads us to the fact that:
[tex]\[ (x \times y) \times z = x \times (y \times z) \][/tex]
This statement is true.
3. Commutativity of Subtraction:
[tex]\( x - y = y - x \)[/tex]
Subtraction of complex numbers is not commutative:
[tex]\[ x - y = (a + bi) - (c + di) = (a - c) + (b - d)i \][/tex]
[tex]\[ y - x = (c + di) - (a + bi) = (c - a) + (d - b)i \][/tex]
Since [tex]\( a - c \neq c - a \)[/tex] and [tex]\( b - d \neq d - b \)[/tex], [tex]\( x - y \neq y - x \)[/tex].
This statement is false.
4. Associativity of Addition:
[tex]\( (x + y) + z = x + (y + z) \)[/tex]
Addition of complex numbers is associative:
[tex]\[ (x + y) + z = ((a + bi) + (c + di)) + (f + gi) = (a + c + f) + (b + d + g)i \][/tex]
[tex]\[ x + (y + z) = (a + bi) + ((c + di) + (f + gi)) = (a + c + f) + (b + d + g)i \][/tex]
Since both sides equate to [tex]\( (a + c + f) + (b + d + g)i \)[/tex], the property holds.
This statement is true.
5. Associativity of Subtraction:
[tex]\( (x - y) - z = x - (y - z) \)[/tex]
Subtraction of complex numbers is not associative:
[tex]\[ (x - y) - z = ((a + bi) - (c + di)) - (f + gi) = (a - c - f) + (b - d - g)i \][/tex]
[tex]\[ x - (y - z) = (a + bi) - ((c + di) - (f + gi)) = (a - c + f) + (b - d + g)i \][/tex]
Therefore, [tex]\( (a - c - f) + (b - d - g)i \)[/tex] does not equal [tex]\( (a - c + f) + (b - d + g)i \)[/tex].
This statement is false.
Based on the analysis:
- True: [tex]\( x + y = y + x \)[/tex], [tex]\( (x \times y) \times z = x \times (y \times z) \)[/tex], [tex]\( (x + y) + z = x + (y + z) \)[/tex]
- False: [tex]\( x - y = y - x \)[/tex], [tex]\( (x - y) - z = x - (y - z) \)[/tex]
