Let * be a binary operation on the set I of all integers, defined by a * b = 3a + 4b - 2. Find the value of 4 * 5.

The binary operation * on R is defined by a * b = 2a + b. Find (2 * 3) * 4.

Let *: R × R → R be a binary operation given by a * b = a + 4b². Then, compute
( - 5) * (2 * 0).

Let be a binary operation on the set Q of all rational numbers given as a * b = (2a - b)² for all a, b ∈ Q. Find 3 * 5 and 5 * 3. Is 3 * 5 = 5 * 3?

Let * be a binary operation on N given by a *b = 1 cm of a and b. Find the value of
20 * 16.

Is * (i) commutative, (ii) associative?

If * be the binary operation on the set Z of all integers defined by a * b (a + 3b²), find 2 * 4

Show that * on Z ⁺ defined by a *b = |a –b| is not a binary operation.

Let * be a binary operation on N, defined by a * b = a^b for all a. b ∈ N.

Show that * is neither commutative nor associative.

Let a * b = 1 cm (a, b) for all values of
a, b ∈ N.

(i) Find (12 * 16).

(ii) Show that * is commutative on N.

(iii) Find the identity element in N.

(iv) Find all invertible elements in N.

Let Q be the set of all positive rational numbers.

(i) Show that the operation * on Q ⁺ defined by is a binary operation.

(ii) Show that * is commutative.

(iii) Show that * is not associative.

Show that the set A = { - 1, 0, 1) is not closed for addition.

Show that * on R –{ - 1}, defined by is neither commutative nor associative.

For all a, b ∈ R, we define a * b = |a – b|.

Show that * is commutative but not associative.

For all a, b ∈ N, we define a * b = a³ + b³.

Show that * is commutative but not associative.

Let X be a nonempty set and * be a binary operation on P(X), the power set of X, defined by A * B = A ∩ B for all A, B ∈ P(X).

(i) Find the identity element in P(X).

(ii) Show that X is the only invertible element in P(X).

A binary operation * on the set (0, 1, 2, 3, 4, 5) is defined as

Show that 0 is the identity for this operation and each element a has an inverse (6 - a)

Define * on N by m * n = 1 cm (m, n). Show that * is a binary operation which is commutative as well as associative.

Define * on Z by a * b = a – b + ab. Show that * is a binary operation on Z which is neither commutative nor associative.

Define * on Z by a * b = a + b - ab. Show that * is a binary operation on Z which is commutative as well as associative.

Consider a binary operation on Q – {1}, defined by a * b = a + b - ab.

(i) Find the identity element in Q – {1}.

(ii) Show that each a ∈ Q - {1} has its inverse.

Let Q₀ be the set of all nonzero rational numbers. Let * be a binary operation on Q₀, defined by for all a, b ∈ Q₀.

(i) Show that * is commutative and associative.

(ii) Find the identity element in Q₀.

(iii) Find the inverse of an element a in Q₀.

On the set Q⁺ of all positive rational numbers, define an operation * on Q⁺ by for all a, b ∈ Q⁺. Show that

(i) * is a binary operation on Q⁺,

(ii) * is commutative,

(iii) * is associative.

Find the identity element in Q⁺ for *. What is the inverse of a ∈ Q⁺?

Let Q⁺ be the set of all positive rational numbers.

(i) Show that the operation * on Q⁺ defined by is a binary operation.

(ii) Show that * is commutative.

(iii) Show that * is not associative.

Let Q be the set of all rational numbers. Define an operation on Q – {-1} by a * b = a + b + ab.

Show that

(i) * is a binary operation on Q – {-1},

(ii) * is Commutative,

(iii) * is associative,

(iv) zero is the identity element in Q – {-1} for *,

Let A = N × N. Define * on A by (a, b) * (c, d) = (a + c, b + d).

Show that

(i) A is closed for *,

(ii) * is commutative,

(iii) * is associative,

(iv) identity element does not exist in A.

Let A = (1, -1, i, -i) be the set of four 4th roots of unity. Prepare the composition table for multiplication on A and show that

(i) A is closed for multiplication,

(ii) multiplication is associative on A,

(iii) multiplication is commutative on A,

(iv) 1 is the multiplicative identity,

(v) every element in A has its multiplicative inverse.