Is ∗ both associative and commutative?Is ∗ commutative but not associative?Is ∗ associative but not commutative?Is ∗ neither commutative nor associative? solution: A two fold activity ∗ on N...
State whether the following statements are true or false. Justify.
For an arbitrary binary operation * on a set N, a * a = a ∀ a ∈ N.If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a solution: (i) Given: * being a paired procedure on N, is...
Let A = N × N and ∗ be the binary operation on A defined by (a, b) ∗ (c, d) = (a + c, b + d) Show that ∗ is commutative and associative. Find the identity element for ∗ on A, if any.
solution: A = N x N and * is a paired activity characterized on A. (a, b) * (c, d) = (a + c, b + d) (c, d) * (a, b) = (c + a, d + b) = (a + c, b + d) The activity * is commutative Once more, ((a, b)...
Find which of the operations given above has identity.
Solution: Let I be the identity. a * I = a – I ≠ aa * I = a2 – I2 ≠ aa * I = a + a I ≠ aa * I = (a – I) 2 ≠ aa * I = aI/4 ≠ a Which is only possible at I = 4 i.e. a * I = aI/4 = a(4)/4 = a 6. a * I...
Let ∗ be a binary operation on the set Q of rational numbers as follows and Find which of the binary operations are commutative and which are associative.
a ∗ b = ab/4a ∗ b = ab2 (v) a ∗ b = abdominal muscle/4 b * a = ba/2 = abdominal muscle/2 a ∗ b = b * a The activity * is commutative. Check for affiliated: (a * b) * c = abdominal muscle/4 * c =...
Let ∗ be a binary operation on the set Q of rational numbers as follows and Find which of the binary operations are commutative and which are associative.
a ∗ b = a + aba ∗ b = (a – b)2 (iii) a ∗ b = a + stomach muscle a ∗ b = a + stomach muscle = a(1 + b) b * a = b + ba = b (1+a) a ∗ b ≠ b * a The activity * isn't commutative Check for cooperative:...
Let ∗ be a binary operation on the set Q of rational numbers as follows and Find which of the binary operations are commutative and which are associative.
a ∗ b = a – ba ∗ b = a2 + b2 Arrangement: (i) a ∗ b = a – b a ∗ b = a – b = - (b – a) = - b * c ≠ b * a (Not commutative) (a * b) * c = (a – b) * c = (a – (b – c) = a – b + c ≠ a * (b *c) (Not...
Let ∗ be the binary operation on N defined by a ∗ b = H.C.F. of a and b. Is ∗ commutative? Is ∗ associative? Does there exist identity for this binary operation on N?
solution: The activity ∗ be the double procedure on N characterized by a ∗ b = H.C.F. of an and b a * b = H.C.F. of an and b = H.C.F. of b and a = b * a Therefore, activity * is commutative. Once...
Is ∗ defined on the set {1, 2, 3, 4, 5} by a ∗ b = L.C.M. of a and b a binary operation? Justify your answer.
solution: The activity ∗ characterized on the set {1, 2, 3, 4, 5} by a ∗ b = L.C.M. of an and b Assume, a = 2 and b = 3 2 * 3 = L.C.M. of 2 and 3 = 6 However, 6 doesn't has a place with the set A....
Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find
Which elements of N are invertible for the operation ∗? (v) Which components of N are invertible for the activity ∗? Just the component 1 in N is invertible for the activity * in light of the fact...
Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find
Is ∗ associative?Find the identity of ∗ in N (iii) Is ∗ affiliated? For a,b, c ∈ N (a ∗ b) * c = (L.C.M. of an and b) * c = L.C.M. of a, b and c a ∗ (b * c) = a * (L.C.M. of b and c) = L.C.M. of a,...
Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find
5 ∗ 7, 20 ∗ 16Is ∗ commutative? solution: (i) 5 ∗ 7 = LCM of 5 and 7 = 35 20 ∗ 16 = LCM of 20 and 16 = 80 (ii) Is ∗ commutative? a ∗ b = L.C.M. of an and b ∗ a = L.C.M. of b and a ∗ b = b ∗ a Along...
Let ∗′ be the binary operation on the set {1, 2, 3, 4, 5} defined by a ∗′ b = H.C.F. of a and b. Is the operation ∗′ same as the operation ∗ defined in Exercise 4 above? Justify your answer.
solution: Let A = {1, 2, 3, 4, 5} and a ∗′ b H.C.F. of an and b. Plot a table qualities, we have Activity ∗′ same as the activity *.
Consider a binary operation ∗ on the set {1, 2, 3, 4, 5} given by the following multiplication table (Table 1.2).
(i) Compute (2 ∗ 3) ∗ 4 and 2 ∗ (3 ∗ 4)Is ∗ commutative? (ii)Compute (2 ∗ 3) ∗ (4 ∗ 5). (Hint: use the following table) solution: (I) Compute (2 ∗ 3) ∗ 4 and 2 ∗ (3 ∗ 4) From table: (2 ∗ 3) = 1 and...
Consider the binary operation ∧ on the set {1, 2, 3, 4, 5} defined by a ∧ b = min {a, b}. Write the operation table of the operation ∧ .
SOLUTION: The paired activity ∧ on the set, say A = {1, 2, 3, 4, 5} characterized by a ∧ b = min {a, b}. the activity table of the activity ∧ as follow:
For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.
On Z+, define a ∗ b = abOn R – {– 1}, define a ∗ b = a/(b+1) (v) On Z+, characterize a ∗ b = abdominal muscle Stage 1: Check for commutative Consider ∗ is commutative, then, at that point a ∗ b = b...
For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.
On Q, define a ∗ b = ab/2On Z+, define a ∗ b = 2ab (iii) On Q, characterize a ∗ b = stomach muscle/2 Stage 1: Check for commutative Consider ∗ is commutative, then, at that point a ∗ b = b * a Which...
For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.
On Z, define a ∗ b = a – bOn Q, define a ∗ b = ab + 1 (i) On Z, characterize a ∗ b = a – b Stage 1: Check for commutative Consider ∗ is commutative, then, at that point a ∗ b = b * a Which implies,...
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.(v)
On Z+, define ∗ by a ∗ b = a On Z+ = {1, 2, 3, 4, 5,… … .} Let a = 2 and b = 1 Therefore, a ∗ b = 2 ∈ Z+ Activity * is a paired procedure on Z+ .
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 a ∗ b = ab2On Z+, define ∗ by a ∗ b = | a – b | (iii) On R, characterize ∗ by a ∗ b = ab2 R = { - ∞, … , - 1, 0, 1, 2,… … , ∞} Let a = 1.2 and b = 2 Accordingly, a ∗ b = ab2 =...
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 Z+, define ∗ by a ∗ b = a – bOn Z+, define ∗ by a ∗ b = ab (i) On Z+ = {1, 2,3 , 4, 5,… … .} Let a = 1 and b = 2 Subsequently, a ∗ b = a – b = 1 – 2 = - 1 ∉ Z+ activity * is certifiably not a...