Question

i) The operation 5 defined by x5y = jln(xy)j where ln x is the natural
logarithm.
ii) The operation 4 defined by x4y = x2+y3.
Also, for those operations which are binary operations, check whether they are
associative and commutative

Answer 1

Answer on Question #79921 – Math – Linear Algebra

Question

i) The operation 5 defined by $x5y = j\ln(xy)j$ where $\ln x$ is the natural logarithm.

ii) The operation 4 defined by $x4y = x2 + y3$.

Also, for those operations which are binary operations, check whether they are associative and commutative

Solution

Associative property:

$$(a * b) * c = a * (b * c);$$

Commutative property:

$$a * b = b * a,$$

where * is binary operation.

i. $x5y = |\ln(xy)|$

Associative property:

$$\begin{array}{l} (x5y)5z = |\ln(|\ln(xy)| \cdot z)| = |\ln(|\ln xy|) + \ln z| \\ x5(y5z) = |\ln(x \cdot |\ln yz|)| = |\ln x + \ln(|\ln yz|)| \\ (x5y)5z \neq x5(y5z) \end{array}$$

— this operation is not associative.

Commutative property:

$$\begin{array}{l} x5y = |\ln xy| \\ y5x = |\ln yx| = |\ln xy|. \\ x5y = y5x \end{array}$$

— this operation is commutative.

ii. $x4y = x^{2} + y^{3}$

Associative property:

$$\begin{array}{l} (x4y)4z = (x^{2} + y^{3})^{2} + z^{3} \\ x4(y4z) = x^{2} + (y^{2} + z^{3})^{3} \\ (x4y)4z \neq x4(y4z) \end{array}$$

— this operation is not associative.

Commutative property:

$$\begin{array}{l} x4y = x^{2} + y^{3} \\ y4x = y^{2} + x^{3} \end{array}$$ $$x 4 y \neq y 4 x$$

— this operation is not commutative.

Answer provided by https://www.AssignmentExpert.com