Commutative Law of Binary Operation

Q.

Which of the following binary operation is not commutative?

1. Addition

2. Subtraction

3. Multiplication

4. None of these

Q.

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

1. False

2. True

Q.

Which of the following binary operation is not commutative?

1. None of these

2. Multiplication

3. Addition

4. Subtraction

Q.

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

1. True

2. False

Q. Let * be the binary operation defined on Q. Which of the following binary operations are commutative?

1. a * b = a – b ∀ a, b ∈ Q
2. a * b = $a^2+b^2$ ∀ a, b ∈ Q
3. a * b = a + ab ∀ a, b ∈ Q
4. a * b =$(a–b{)}^2$ ∀ a, b ∈ Q

Q.

Which of the following binary operation is not commutative?

1. Addition

2. Multiplication

3. Subtraction

4. None of these

Q. Let * be the binary operation defined on Q. Which of the following binary operations are commutative?

1. a * b = a – b ∀ a, b ∈ Q
2. a * b = $a^2+b^2$ ∀ a, b ∈ Q
3. a * b = a + ab ∀ a, b ∈ Q
4. a * b =$(a–b{)}^2$ ∀ a, b ∈ Q

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. 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. Let * be a binary operation on the set Q of rational numbers as follows: (i) a * b = a − b (ii) a * b = a 2 + b 2 (iii) a * b = a + ab (iv) a * b = ( a − b ) 2 (v) (vi) a * b = ab 2 Find which of the binary operations are commutative and which are associative.