Question
Check the commutativity and associativity of the following binary operations:
$'⊙'$ on $Q$ defined by $a ⊙ b = a^2 + b^2$ for all $a, b \in Q.$
$'⊙'$ on $Q$ defined by $a ⊙ b = a^2 + b^2$ for all $a, b \in Q.$
✓
Answer
Commutativity: Let $\text{a, b}\in\text{Q}$ then,
$a ⊙ b = a^2 + b^2 = b^2 + a^2 = b ⊙ a$
Therefore, $a ⊙ b = b ⊙ a$, $\forall\ \text{a, b}\in\text{Q}$
Thus, $⊙$ is commuatative on $Q.$
Associativity: Let $\text{a, b, c}\in\text{Q.}$
$a ⊙ (b ⊙ c) = a ⊙ (b^2 + c^2) $
$= ab2 + (b^2 + c^2)^2 = ab^2 + b^4 + c^4 + 2b^2c^2 (a ⊙ b) ⊙ c $
$= (a^2 + b^2) ⊙ c = (a^2 + b^2)^2 + c^2 $
$=a^4 + b^4 + 2a^2b^2 + c^2$
Therefore,$\text{a}\odot\text{b}\odot\text{c}\neq\text{a}\odot\text{b}\odot\text{c}$
Thus, $⊙$ is not associative on $Q.$
$a ⊙ b = a^2 + b^2 = b^2 + a^2 = b ⊙ a$
Therefore, $a ⊙ b = b ⊙ a$, $\forall\ \text{a, b}\in\text{Q}$
Thus, $⊙$ is commuatative on $Q.$
Associativity: Let $\text{a, b, c}\in\text{Q.}$
$a ⊙ (b ⊙ c) = a ⊙ (b^2 + c^2) $
$= ab2 + (b^2 + c^2)^2 = ab^2 + b^4 + c^4 + 2b^2c^2 (a ⊙ b) ⊙ c $
$= (a^2 + b^2) ⊙ c = (a^2 + b^2)^2 + c^2 $
$=a^4 + b^4 + 2a^2b^2 + c^2$
Therefore,$\text{a}\odot\text{b}\odot\text{c}\neq\text{a}\odot\text{b}\odot\text{c}$
Thus, $⊙$ is not associative on $Q.$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Obtain the differential euqation by eliminating the arbitrary constants from the following :
$y=c_1 e^{3 x}+c_2 e^{2 x}$
→$y=c_1 e^{3 x}+c_2 e^{2 x}$
Find the probability distribution of the number of doublets in 4 throws of a pair of dice.
Also, Find the mean and variance of this distribution.
Also, Find the mean and variance of this distribution.
If $\text{x}\begin{bmatrix}2\\3 \end{bmatrix}+\text{y}\begin{bmatrix}-1\\1 \end{bmatrix}=\begin{bmatrix}10\\5 \end{bmatrix},$ find the value of x.
→If in parallelogram $A B C D$, diagonal vectors are $\overline{A C}=2 \hat{i}+3 \hat{j}+4 \hat{k}$ and $\overline{B D}=$
→$-6 \hat{i}+7 \hat{j}-2 \hat{k}$, then find the adjacent side vectors $\overline{A B}$ and $\overline{A D}$
Verify Rolle’s theorem for the following functions.
f(x) = sin x + cos x + 7, x ∈ [0, 2π]
Find the inverse of the following matrices$\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right]$
→Integrate the following integrals:
$\int\cos3\text{x}\cos4\text{x dx}$
→$\int\cos3\text{x}\cos4\text{x dx}$
Express the following circuits in the symbolic form of logic and writ the input-output table.
Prove that $\cos^{-1}\frac{4}{5}+\cos^{-1}\frac{12}{13}=\cos^{-1}\frac{33}{65}$
→Show that : $\sin ^{-1}\left(\frac{8}{17}\right)+\sin ^{-1}\left(\frac{3}{5}\right)=\sin ^{-1}\left(\frac{77}{85}\right)$
→