Let * be a binary operation on Z defined by a * b = a + b - 4 for… - Vidyadip

Question

Let * be a binary operation on Z defined by a * b = a + b - 4 for all a, b ∈ Z.
Show that '*' is both commutative and associative.

✓

Answer

Commutativity: Let $\text{a, b}\in\text{Z}.$ Then,
a * b = a + b - 4
= b + a - 4
= b * a
Therefore,
a * b = b * a, $\forall\ \text{a, b}\in\text{Z}$
Thus, * is commutative on Z.
Associativity: Let $\text{a, b, c}\in\text{Z}.$ Then,
a * (b * c) = a * (b + c - 4)
= a + b + c - 4 - 4
= a + b + c - 8
(a * b) * c = (a + b - 4) * c
= a + b - 4 + c - 4
= a + b + c - 8
Therefore, a * (b * c) = (a * b) * c, $\forall\ \text{a, b, c}\in\text{Z}.$
Thus, * is associative on Z.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Binary Operations→Binary Operations — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Prove that $\cot^{-1}\bigg(\frac{\sqrt{1 +\sin\text{x}} + \sqrt{1 -\sin\text{x}}}{\sqrt{1 + \sin\text{x}} - \sqrt{1 - \sin\text{x}}} \bigg)= \frac{\text{x}}{2}; \text{x}\in\bigg(0,\frac{\pi}{4}\bigg).$

Differentiate the following functions with respect to x:
$2^{\text{x}^3}$

Find the adjoint of the following matrices: $\begin{bmatrix}-3 & 5 \\ 2 & 4 \end{bmatrix}$Verify that (adjoint A) A = |A|I = A (adjoint A) for the above matrices.

Let $\text{A}=\begin{bmatrix}-1&0&2\\3&1&4 \end{bmatrix},\text{ B}=\begin{bmatrix}0&-2&5\\1&-3&1 \end{bmatrix}$ and $\text{C}=\begin{bmatrix}1&-5&2\\6&0&-4 \end{bmatrix}.$ Compute 2A - 3B + 4C.

Evaluate the following integrals:
$\int\limits^\pi_0\cos\text{x}|\cos\text{x}|\text{dx}$

Two cards are drawn successively without replacement from a well-shuffled deck of 52 cards. Find the probability of exactly one ace.

A circular disc of radius 3cm is being heated. Due to expansion, its radius increases at the rate of 0.05cm/ sec. Find the rate at which its area is increasing when radius is 3.2cm.

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

Evaluate the following integrals:
$\int\text{x}\frac{\tan^{-1}\text{x}^2}{1+\text{x}^4}\text{ dx}$

Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\text{x}^2\cos2\text{x}\text{ dx}$