Question

Logical Reasoning Copy and complete the table. Use the results in each row to decide whether subtraction and division are commutative or associative operations. Explain your reasoning.

Short answer

Subtraction & division are neither commutative nor associative.

Step-by-step solution

Step-1 – Commutative and Associative Properties of Addition

Commutative Property of Addition:

Property: a + b = b + a

If you add two real numbers in any order, the sum will always be the same.

Example: 5 + 2 = 2 + 5 = 7

Associative property of Addition

(a + b) + c = a + (b + c)

If you are adding three real numbers, the sum is always the same regardless of their grouping.

Example: (11+12) + 13 = 11 + (12+13) = 36

Step-2 – Commutative and Associative Properties of Multiplication

Commutative Property of Multiplication:

Property: $\text{a} · \text{b} =\text{b} · \text{a}$ or $\text{a} · \text{b} · \text{c}=\text{b} · \text{c} · \text{a}$

If you multiply two or three real numbers in any order, their product will always be the same.

Example: $\text{12} · \text{4} =\text{4} · \text{12} = 48$ or $\text{5} · \text{6} · \text{4}=\text{6} · \text{4} · \text{5} \text{=} \text{120}$

Associative property of Multiplication

Property: $\text{(a} · \text{b)} · \text{c}=\text{a} · ( \text{b} · \text{c} )$

If you are multiplying three real numbers, the product is always the same regardless of their grouping.

Example: $\text{(6} · \text{4)} · \text{3}=\text{6} · ( \text{4} · \text{3} ) = 72$

In Real number Addition and multiplication only obeys the Commutative and Associative Property.

Subtraction & division are neither commutative nor associative.

Step-3 – Copy and complete the table. Use the results in each row to decide whether subtraction and division are commutative or associative operations. Explain your reasoning.

Subtraction & division are neither commutative nor associative.

Explanation:

$\begin{array}{l}8-3=5;3-8=-58-3\ne 3-8\\ 10÷5=25÷10=\frac{1}{2}10÷5\ne 5÷10\\ \left(15-9\right)-4=6-4=215-\left(9-4\right)=15-5=10\left(15-9\right)-4\ne 15-\left(9-4\right)\\ \left(48÷6\right)÷2=8÷2=448÷\left(6÷2\right)=48÷3=16\left(48÷6\right)÷2\ne 48÷\left(6÷2\right)\end{array}$

We can conclude that:

$8-3\ne 3-8$Hence subtraction is not commutative.

$10 ÷ 5 \ne 5 ÷ 10$Hence division is not commutative.

$\left(15-9\right)-4\ne 15-\left(9-4\right)$Hence subtraction is not associative.

$\left(48÷6\right)÷2 \ne 48÷\left(6÷2\right)$ Hence division is not associative.