Commutative Law of Binary Operation

Q.

If A is a subset of B , then prove that A union B is B .

Q.

If every element of a group $G$ is its own inverse, then $G$

Q.

If $a+b+c+d=4$ then find the value of $\frac{1}{\left(1-a\right)\left(1-b\right)\left(1-c\right)}+\frac{1}{\left(1-b\right)\left(1-c\right)\left(1-d\right)}+\frac{1}{\left(1-c\right)\left(1-d\right)\left(1-a\right)}+\frac{1}{\left(1-d\right)\left(1-a\right)\left(1-b\right)}$

Q.

The relation $a\ast b=(a-b{)}^2$ , is commutative -

1. True

2. False

Q.

Let $\ast$ be the binary operation on N given by $a\ast b=LCM$ of a and b.
(i) Find $5\ast 7,20\ast 16$

(ii)Is $\ast$ commutative?

(iii)Is $\ast$ associative?

(iv) Find the identity of $\ast$ in N

(v)Which elements of N are invertible for the operation $\ast$ ?

Q. Consider a binary operation * on N defined as a * b = a 3 + b 3 . Choose the correct answer. (A) Is * both associative and commutative? (B) Is * commutative but not associative? (C) Is * associative but not commutative? (D) Is * neither commutative nor associative?

Q.

The default parameter passing mechanism

1. call by value

2. call by reference

3. call by value result

4. none of these

Q.

Why managing devops via chat is called chatops?

Q.

Which of the following binary operation is not commutative?

1. None of these

2. Multiplication

3. Addition

4. Subtraction

Q. If $f\left(x\right)=|x^2-5x+6|$, then which among the following is/are CORRECT?

1. $f^{\prime }\left(x\right)=2x-5,\text{if}x>3\text{or}x<2$
2. $f^{\prime }\left(x\right)=2x-5$ for all $x$
3. $f^{\prime }\left(x\right)=5-2x,\text{if}2<x<3$
4. $f^{\prime }\left(x\right)=2x-5,\text{if}x\ge 3\text{or}x\le 2$

Q.

Let $\ast$ be a binary operation on the set Q of rational number as follows:
$\left(i\right)a\ast b=a-b$

$\left(ii\right)a\ast b=a^2+b^2$

$\left(iii\right)a\ast b=a+ab$

$\left(iv\right)a\ast b=(a-b{)}^2$

$\left(v\right)a\ast b=\frac{ab}{4}$

$\left(vi\right)a\ast b=a{b}^2$
Find which of the binary operation are commutative and which are associative?

Q. Determine the given below binary operation on the set $R$ is associative or commutative.

(i) $a\ast b=1\forall a,b\in R$

Q. Subtraction of integers is
(a) commutative but no associative
(b) commutative and associative
(c) associative but not commutative
(d) neither commutative nor associative

Q.

We can say that composition of functions is commutative since fog (x) = gof (x).

1. True

2. False

Q.

Let $\mathrm{h}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right)-(\mathrm{f}{\left(\mathrm{x}\right))}^2+(\mathrm{f}{\left(\mathrm{x}\right))}^3$ for every real number $\mathrm{x}$. Then :

1. $\mathrm{h}$ is increasing whenever $\mathrm{f}$ is increasing

2. $\mathrm{h}$ is increasing whenever $\mathrm{f}$ is decreasing

3. $\mathrm{h}$ is decreasing whenever $\mathrm{f}$ is increasing

4. nothing can be said in general

Q.

Consider the binary operations*: R ×R → and o: R × R → R defined as and a o b = a, &mnForE;a, b ∈ R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that &mnForE;a, b, c ∈ R, a*(b o c) = (a * b) o (a * c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.

Q.

Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b $-$ ab, for all a, b $\in$ S.
Prove that:
(i) * is a binary operation on S
(ii) * is commutative as well as associative. [CBSE 2014]

Q. The law a + b = b + a is called
(a) closure law
(b) associative law
(c) commutative law
(d) distributive law

Q. If A and B are square matrices of the same order, explain, why in general
(i) (A + B)² ≠ A² + 2AB + B²
(ii) (A − B)² ≠ A² − 2AB + B²
(iii) (A + B) (A − B) ≠ A² − B².

Q. The number of commutative binary operations that can be defined on a set of 2 elements is
(a) 8
(b) 6
(c) 4
(d) 2

Q.

For given binary operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative.
(ii)On Q, define $a\ast b=ab+1$

Q. On Z an operation * is defined by a * b = a² + b² for all a, b ∈ Z. The operation * on Z is
(a) commutative and associative
(b) associative but not commutative
(c) not associative
(d) not a binary operation

Q.

For each binary operation ∗ defined below, determine whether ∗ is commutative or associative.

(iii) On $Q$, define $a\ast b=\frac{ab}{2}$

Q. Consider the binary operations*: R × R → and o: R × R → R defined as and a o b = a , &mnForE; a , b ∈ R . Show that * is commutative but not associative, o is associative but not commutative. Further, show that &mnForE; a , b , c ∈ R , a *( b o c ) = ( a * b ) o ( a * c ). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.

Q.

Consider a binary operation * on N defined as a * b = a³ + b³. Choose the correct answer.

(A) Is * both associative and commutative?

(B) Is * commutative but not associative?

(C) Is * associative but not commutative?

(D) Is * neither commutative nor associative?

Q. State whether the given below statement is true or false. Justify.

If $\ast$ is a commutative binary operation on $N,$ then $a\ast (b\ast c)=(c\ast b)\ast a$

Q. Consider the binary operations

$\ast :R\times R\to Rando:R\times R\to R$ defined as $a\ast b=|a-b|$ and $a\circ b=a,\forall a,b\in R.$Show that * is commutative but not associative, o is associative but not commutative. Further, show that $\forall a,b,c\in R,a\ast (b\circ c)=(a\ast b)\circ (a\ast c)$. (If it is so, we say that the operation * distributes over the operation o). Does o distribute over? Justify your answer.

Q.

For given binary operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative.
(vi) On R-{-1}, define $a\ast b=\frac{a}{b+1}$

Q. The binary operation * defined on N by

a * b = a + b + ab for all a, b ∈ N is

(a) commutative only
(b) associative only
(c) commutative and associative both
(d) none of these

Q.

Let A and B be sets. If A ∩ X = B ∩ X = Φ and A ∪ X = B ∪ X for some set X, show that A = B.

(Hints A = A ∩ (A ∪ X), B = B ∩ (B ∪ X) and use distributive law)