Cummutative Law of Binary Operation

Q. is a binary operation which is not commutative.

1. Addition
2. Subtraction
3. Multiplication

Q.

For each binary operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative.
(i) On Z, define a $\ast$ b =a-b

(ii) On Q, define $a\ast b=ab+1$

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

(iv) On $Z^{+}$, define $a\ast b=2^{ab}$

(v) On $Z^{+}$, define $a\ast b=a^b$

(vi) On (R-{-1} , define $a\ast b=\frac{a}{b+1}$

Q. The maximum number of nodes on level $5$ of a binary tree.

1. $32$
2. $16$
3. $64$
4. $63$

Q.

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

1. True

2. False

Q.

Consider a binary operation $\ast$ on N defind as $a\ast b=a^3+b^3$. Choose the correct answer.
$\left(a\right)\ast$ both associative and commutative
$\left(b\right)\ast$ commutative but not associative
$\left(c\right)\ast$ is associative but not commutative
$\left(d\right)\ast$ is neither commutative nor associative

Q. If ${}^{\prime }{\ast }^{\prime }$ is a binary operation on the set R of real numbers defined by $a\ast b=a+b-2$, find the identity element for the binary operation ${}^{\prime }{\ast }^{\prime }$.

Q.

State whether the following statement are true or false. Justify
(i) for an arbitrary binary operation $\ast$ on a set N, $a\ast a=a\forall a\in N$.

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

Q. If $r,s$ and $t$ are prime numbers and $p,q$ are positive integers such that the LCM of $p,q$ is $r^2{t}^4{s}^2$ then the number of ordered pair $(p,q)$ is

1. 254
2. 252
3. 225
4. 224

Q.

Which of the following is the identity pair of addition and multiplication in order?

1. (0, 1)

2. (1, 1)

3. (1, 0)

4. (0, 0)

Q. Let $\ast$ be the binary operation on $N$ given by $a\ast b=L.C.M$ of $a$ and $b$. Find
(i) $5\ast 7,20\ast 16$ (ii) Is $\ast$ commutative?

Q. How/Why do $nPr$ and $nCr$ work?

Q.

Which of the following binary operation is not commutative?

1. Addition

2. Multiplication

3. Subtraction

4. None of these