1. 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. (i) On Z+, define * by a * b = a b (ii) On Z+, define * by a * b = ab (iii) On R, define * by a * b = ab2 (iv) On Z+, define * by a * b = | a b| (v) On Z+, define * by a * b = a 2. For each operation * defined below, determine whether * is binary, 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 = 2 ab (iv) On Z+, define a * b = 2ab (v) On Z+, define a * b = ab (vi) On R { 1}, define a * b = 1 a b + 3. 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 . 4. Consider a binary
Question 1: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. (i) On Z+, define * by a * b = a – b (ii) On Z+, define * by a * b = ab (iii) On R, define * by a * b = ab2 (iv) On Z+, define * by a * b = | a – b| (v) On Z+, define * by a * b = a
Question 2:For each operation * defined below, determine whether * is binary, 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 = 2 ab (iv) On Z+, define a * b = 2ab (v) On Z+, define a * b = ab (vi) On R – {– 1}, define a * b = 1 a b +
Question 3:Consider the binary operation Ù on the set {1, 2, 3, 4, 5} defined by a Ù b = min {a, b}. Write the operation table of the operation Ù .
Question 4: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) (ii) Is * commutative? (iii) Compute (2 * 3) * (4 * 5). (Hint: use the following table) Table 1.2
Question 5: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.
Question 6:Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find (i) 5 * 7, 20 * 16 (ii) Is * commutative? (iii) Is * associative? (iv) Find the identity of * in N (v) Which elements of N are invertible for the operation *?
Question 7: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.
Question 8: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?
Question 9:Let * be a binary operation on the set Q of rational numbers as follows: (i) a * b = a – b (ii) a * b = a2 + b2 (iii) a * b = a + ab (iv) a * b = (a – b)2 (v) a * b = 4 ab (vi) a * b = ab2 Find which of the binary operations are commutative and which are associative.
Question 10:Find which of the operations given above has identity.
Question 11: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.
Question 12:State whether the following statements are true or false. Justify. (i) For an arbitrary binary operation * on a set N, a * a = a "a Î N. (ii) If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a
Question 13:Consider a binary operation * on N defined as a * b = a3 + b3. Choose the correct answer. (A) Is * both associative and commutative? (B) Is * commutative but not associative? (C) Is * associative but not commutative? (D) Is * neither commutative nor associative?
