Associative Law of Binary Operation

Q.

Two finite sets have $m$ and $n$ element respectively The total number of subsets of the first set is $112$ more than the total number of subsets of the second set The value of $m$ and $n$ respectively are

1. $5,2$

2. $4,7$

3. $7,4$

4. $2,5$

Q. Dear sir
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I need to clear my concepts on ‘number of divisors of a natural number N’ from chapter Permutations and combinations in Maths.
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Q.

Let * be any binary operation on the set R defined by a * b = a + b – ab, then the binary operation * is

1. Commutative and associative

2. Commutative but not associative

3. Associative

4. Commutative but not associative

Q. If $\left[\begin{array}{ccc}9& -1& 4\\ -2& 1& 3\end{array}\right]=A+\left[\begin{array}{ccc}1& 2& -1\\ 0& 4& 9\end{array}\right]$, then find matrix A.

Q. How to check if some given definition is binary operations or not?

Q.

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

(i) On Z, define a * b = a − b

(ii) On Q, define a * b = ab + 1

(iii) On Q, define a * b

(iv) On Z⁺, define a * b = 2^ab

(v) On Z⁺, define a * b = a^b

(vi) On R − {−1}, define

Q.

The law a + b = b + a is called

1. Closure law

2. Associative law

3. Commutative law

4. Distributive law

Q. If $A=\left[\begin{array}{ccc}1& 0& -3\\ 2& 1& 3\\ 0& 1& 1\end{array}\right]$, then verify that A² + A = A(A + I), where I is the identity matrix.

Q.

If * and ^O are two binary operations defined by a * b = a + b and a ^O b = ab, then

1. * is distributive on ^O.

2. a ^O (b * c) = ac + bc

3. (a * b) ^O c = ab + ac

4. ^O is distributive on *.

Q.

Which of the following binary operations defined on the set of real numbers is not associative?

1. Addition

2. Subtraction

3. Multiplication

4. Division

Q.

In class 11 math if we study chapter SETS, PERMUTATION AND COMBINATION AND PROBABLITY FIRST then their is also a need of another concept of any other chapter of class 11 for these all chapter?

Q. Examine whether the operation $\ast$ defined on $R$ by $a\ast b=ab+1$ is
(i) a binary or not.
(ii) if a binary operation, is it associative or not ?

Q. Let * be a binary operation defined on Q. Which of the following binary operation(s) is/are associative?

1. a * b = a – b for a, b ∈ Q.
2. a * b = $\frac{ab}{4}$ for a, b ∈ Q.
3. a * b = a – b + ab for a, b ∈ Q
4. a * b = $a{b}^2$ for a, b ∈ Q

Q.

If A = N x N and * be any binary operation on A defined by

(a, b) * (c, d) = (a + c, b + d), then the binary operation is

1. Associative

2. Commutative

3. Commutative and associative

4. Commutative but not associative

Q.

Let $\ast$ be a binary operation on the Q of rational number as follows:
$\left(v\right)a\ast b=\frac{ab}{4}$
Find which of the binary operation are commutative and which are associative?

Q. Write the inverse of 5 under multiplication modulo 11.

Q. For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A (BC):
(i) $A=\left[\begin{array}{ccc}1& 2& 0\\ -1& 0& 1\end{array}\right],B=\left[\begin{array}{cc}1& 0\\ -1& 2\\ 0& 3\end{array}\right]\mathrm{and}C=\left[\begin{array}{c}1\\ -1\end{array}\right]$

(ii) $A=\left[\begin{array}{ccc}4& 2& 3\\ 1& 1& 2\\ 3& 0& 1\end{array}\right],B=\left[\begin{array}{ccc}1& -1& 1\\ 0& 1& 2\\ 2& -1& 1\end{array}\right]\mathrm{and}C=\left[\begin{array}{ccc}1& 2& -1\\ 3& 0& 1\\ 0& 0& 1\end{array}\right]$.

Q.

Let $\ast$ be a binary operation on the set Q of rational number as follows:
$\left(i\right)a\ast b=a{b}^2$
Find which of the binary operation are commutative and which are associative?

Q. $\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+a& a\\ 1& 1& 1+a\end{array}\right|=a^3+3a^2$

Q. Let $A=R\times R$ and $\ast$ be a binary operation on A defined by $(a,b)\ast (c,d)=(a+c,b+d)$. Show that $\ast$ on A. Also, find the inverse of every element $(a,b)\in A$.

Q. Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative.

Q. $\left|\begin{array}{ccc}1& x& x^2\\ x^2& 1& x\\ x& x^2& 1\end{array}\right|={\left(1-x^3\right)}^2$

Q.

Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table.

(i) Compute (2 * 3) * 4 and 2 * (3 * 4)

(ii) Is * commutative?

(iii) Compute (2 * 3) * (4 * 5).

(Hint: use the following table)

*	1	2	3	4	5
1	1	1	1	1	1
2	1	2	1	2	1
3	1	1	3	1	1
4	1	2	1	4	1
5	1	1	1	1	5

Q. Write the composition table for the binary operation multiplication modulo 10 (×₁₀) on the set S = {2, 4, 6, 8}.

Q. An operation * is defined on the set Z of non-zero integers by $a*b=\frac{a}{b}$ for all a, b ∈ Z. Then the property satisfied is
(a) closure
(b) commutative
(c) associative
(d) none of these

Q. Consider a binary operation ∗ on N defined as a ∗ b =$a^3+b^3$ . Which of the following statement(s) is/are correct?

1. ∗ is both associative and commutative
2. ∗ is commutative but not associative
3. ∗ is associative but not commutative
4. ∗ is neither commutative nor associative

Q. check weather a*b=(a-b)^2 is commutative or assosiative both

Q. Write the composition table for the binary operation ×₅ (multiplication modulo 5) on the set S = {0, 1, 2, 3, 4}.

Q. On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b − 4. Prove that * is neither commutative nor associative on Z.

Q. Mark the correct alternative in the following question:

$$\begin{gathered} \mathrm{If}\mathrm{P}\left(B\right)=\frac{3}{5},\mathrm{P}\left(A|B\right)=\frac{1}{2}\mathrm{and}\mathrm{P}\left(A\cup B\right)=\frac{4}{5},\mathrm{then}\mathrm{P}\left(\overline{A\cup B}\right)+\mathrm{P}\left(\overline{A}\cup B\right)= \\ \left(\mathrm{a}\right)\frac{1}{5}\left(\mathrm{b}\right)\frac{4}{5}\left(\mathrm{c}\right)\frac{1}{2}\left(\mathrm{d}\right)1 \end{gathered}$$