let $x = a + bi$ and $y = c + di$ and $z = f + gi$
which statements are true? check all of the boxes that apply

$x + y = y + x$
$(x \times y) \times z = x \times (y \times z)$
$x - y = y - x$
$(x + y) + z = x + (y + z)$
$(x - y) - z = x - (y - z)$

Question

let $x = a + bi$ and $y = c + di$ and $z = f + gi$
which statements are true? check all of the boxes that apply

$x + y = y + x$
  $(x \times y) \times z = x \times (y \times z)$
  $x - y = y - x$
  $(x + y) + z = x + (y + z)$
  $(x - y) - z = x - (y - z)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Complex Numbers" ], "new_concepts": [ "Complex Number Properties", "Algebraic Field Properties" ], "current_concepts": [ "Complex Numbers", "Complex Number Properties", "Algebraic Field Properties" ] } ## Evaluate commutativity and associativity of addition and multiplication $$ \begin{aligned} x + y &= (a + c) + (b + d)i = (c + a) + (d + b)i = y + x \quad \text{(True)} \ (x \times y) \times z &= x \times (y \times z) \quad \text{(True, multiplication is associative)} \ (x + y) + z &= x + (y + z) \quad \text{(True, addition is associative)} \end{aligned} $$ ## Evaluate subtraction properties $$ \begin{aligned} x - y &= (a - c) + (b - d)i \ y - x &= (c - a) + (d - b)i eq x - y \quad \text{(False)} \ (x - y) - z &= x - y - z \ x - (y - z) &= x - y + z eq (x - y) - z \quad \text{(False)} \end{aligned} $$

(A) $x + y = y + x$

(B) $(x \times y) \times z = x \times (y \times z)$

(C) $x - y = y - x$

(D) $(x + y) + z = x + (y + z)$

(E) $(x - y) - z = x - (y - z)$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Complex Numbers" ] }

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Question

Explanation:

Evaluate commutativity and associativity of addition and multiplication

Evaluate subtraction properties

Answer: