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The Law A + B = B + A is Called - Mathematics

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प्रश्न

The law a + b = b + a is called _________________ .

विकल्प

  • closure law

  • associative law

  • commutative law

  • distributive law

MCQ

उत्तर

The law a + b = b + a is called commutative law.

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अध्याय 3: Binary Operations - Exercise 3.7 [पृष्ठ ३८]

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आरडी शर्मा Mathematics [English] Class 12
अध्याय 3 Binary Operations
Exercise 3.7 | Q 17 | पृष्ठ ३८

संबंधित प्रश्न

Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

On R, define * by ab2


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

On Z+, define ab


Let * be a binary operation on the set of rational numbers as follows:

(i) − 

(ii) a2 + b2

(iii) ab 

(iv) = (− b)2

(v) a * b = ab/4

(vi) ab2

Find which of the binary operations are commutative and which are associative.


State whether the following statements are true or false. Justify.

If * is a commutative binary operation on N, then * (c) = (b) * a


Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.

 On Z+, define * by a * b = a

Here, Z+ denotes the set of all non-negative integers.


Let * be a binary operation on the set I of integers, defined by a * b = 2a + b − 3. Find the value of 3 * 4.


Let S = {abc}. Find the total number of binary operations on S.


Find the total number of binary operations on {ab}.


Determine which of the following binary operations are associative and which are commutative : * on Q defined by \[a * b = \frac{a + b}{2} \text{ for all a, b } \in Q\] ?


Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = ab2 for all ab ∈ Q ?


 Check the commutativity and associativity of the following binary operation'*' on Q defined by a * b = ab + 1 for all a, b ∈ Q ?


Check the commutativity and associativity of the following binary operation  '*' on Z defined by a * b = a + b − ab for all a, b ∈ Z ?


Check the commutativity and associativity of the following binary operation '*' on N defined by a * b = gcd(a, b) for all a, b ∈ N ?


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 * is commutative as well as associative ?


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 ?


Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Find the identity element in Q − {−1} ?


Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}.


Construct the composition table for ×5 on Z5 = {0, 1, 2, 3, 4}.


Consider the binary operation 'o' defined by the following tables on set S = {a, bcd}.

o  a b c d
a a a a a
b a b c d
c a c d b
d a d b c

Show that the binary operation is commutative and associative. Write down the identities and list the inverse of elements.


For the binary operation multiplication modulo 10 (×10) defined on the set S = {1, 3, 7, 9}, write the inverse of 3.


On the power set P of a non-empty set A, we define an operation ∆ by

\[X ∆ Y = \left( \overline{X} \cap Y \right) \cup \left( X \cap \overline{Y} \right)\]

Then which are of the following statements is true about ∆.


Q+ denote the set of all positive rational numbers. If the binary operation a ⊙ on Q+ is defined as \[a \odot = \frac{ab}{2}\] ,then the inverse of 3 is __________ .


If the binary operation ⊙ is defined on the set Q+ of all positive rational numbers by \[a \odot b = \frac{ab}{4} . \text{ Then }, 3 \odot \left( \frac{1}{5} \odot \frac{1}{2} \right)\] is equal to __________ .


Which of the following is true ?


The binary operation * defined on N by a * b = a + b + ab for all a, b N is ________________ .


On the set Q+ of all positive rational numbers a binary operation * is defined by \[a * b = \frac{ab}{2} \text{ for all, a, b }\in Q^+\]. The inverse of 8 is _________ .


Let * be a binary operation defined on Q+ by the rule

\[a * b = \frac{ab}{3} \text{ for all a, b } \in Q^+\] The inverse of 4 * 6 is ___________ .


Let * be an operation defined as *: R × R ⟶ R, a * b = 2a + b, a, b ∈ R. Check if * is a binary operation. If yes, find if it is associative too.


Let A = `((1, 0, 1, 0),(0, 1, 0, 1),(1, 0, 0, 1))`, B = `((0, 1, 0, 1),(1, 0, 1, 0),(1, 0, 0, 1))`, C = `((1, 1, 0, 1),(0, 1, 1, 0),(1, 1, 1, 1))` be any three boolean matrices of the same type. Find (A v B) ∧ C


Let M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}` and let * be the matrix multiplication. Determine whether M is closed under * . If so, examine the existence of identity, existence of inverse properties for the operation * on M


Choose the correct alternative:

Subtraction is not a binary operation in


In the set N of natural numbers, define the binary operation * by m * n = g.c.d (m, n), m, n ∈ N. Is the operation * commutative and associative?


Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = a – b + ab for a, b ∈ Q


Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

a * b = a2 + b2 ∀ a, b ∈ Q


A binary operation on a set has always the identity element.


The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is ____________.


The identity element for the binary operation * defined on Q – {0} as a * b = `"ab"/2 AA  "a, b" in "Q" - {0}` is ____________.


Consider the binary operation * on Q defind by a * b = a + 12b + ab for a, b ∈ Q. Find 2 * `1/3`.


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