Prob. Let (A, ) be a semigroup. Show that for a, b, c ∈ A, if ac = ca and bc = cb, then (ab)c = c(a*b). Solution:

Prob. Let ({a, b}, * ) be a semigroup where aa =b. Show that- ab=b*a. Sol. Given ({a*b}, *) is a semigroup And a*a = b. Now ab = a(aa) (∵ aa=b) ab =(aa)*a (by associative law) ab =ba (∵ a*a=b)

Prob. Show that the set I of all integers (…, -4, -3, -2, -1, 0, 1, 2, 3, 4,…}. Is a group with respect to the operation of addition of integers? Sol. 1) Closure Property:2+2 = 4;2-2=0;6+4=10’4-6=-2;We know that addition of two integers is also in integer.i.e, a + b ∈ I, ∀ a, b […]

Group A non-empty set G of some elements (a, b, c, etc.), with one or more operations is known as a group. A set needed to be satisfied following properties to become a group: 1) Closure Property:a.b ∈ G , ∀ a, b ∈ G 2) Associative Property:(a . b) . c = a . […]

Algebraic Structures G -> a non-empty set. G with one or more binary operations is known as algebraic structures. For examples 1) (G, ) , where ‘’ is an binary operation on Set/Group ‘G’. Than (G,*) is an algebraic group. 2) (N, +), where ‘+’ is an binary operation on Set/Group ‘N’,set of natural numbers. […]

SET Set is a collection of definite well defined objects. Set is denoted by capital letter. For example: A = {a, b, c, d, e} BINARY OPERATIONS ON A SET Let, G à a non-empty set GXG = {(a ,b): a ∈ G, b ∈ G }. Above line is read as: G cross G equal to (a, b) such that ‘a’ […]

Prob. Show that-(P∩Q)X(R∩S) = (PXR)∩(QXS)For some arbitrary sets P, Q, R and S Solution:Consider(x,y)(x,y)∈(P∩Q)X(R∩S)x∈(P∩Q) ∧ y∈(R∩S)(x∈P and x∈Q) ∧ (y∈R and y∈S)(x∈P ∧ y∈R) and (x∈Q ∧ y∈S)(x,y)∈(P∧R) and (x,y)∈Q∧S)(x,y)∈((P∧R) and (Q∧S))(x,y)∈((P×R) ∩ (Q×S)) (PXR)∩(QXS)

Prob. If A, B, C, D are any four sets then prove that –(A∩B)X(C∩D) = (AXC)∩(BXD) Solution:Consider(x,y)(x,y)∈(A∩B)×(C∩D)x∈(A∩B) ∧ y∈(C∩D)(x∈A and x∈B) ∧ (y∈C and y∈D)(x∈A ∧ y∈C) and (x∈B ∧ y∈D)(x,y)∈(A∧C) and (x,y)∈(B∧D)(x,y)∈((A∧C) and (B∧D))(x,y)∈((A×C) ∩ (B×D)) (A×C) ∩ (B×D)

Prob. Prove that-A∩(B∪C) = (A∩B) ∪ (A∩C) Solution:Let x ∈ A ∩ (B U C).Then x ∈ A and x ∈ (B U C).(x ∈ A and x ∈ B) or (x ∈ A and x ∈ c).x ∈ (A and B) or x ∈ ( A and c). x ∈ (A ∩ B) U […]

Prob. Let A,B,C be any three sets, then prove that-AX(B∩C) = (AXB) ∩ (AXC) Solution:(x,y) ∈ Ax(B∩C)x∈A and (y∈(B∩C))x∈A and (y∈B and y∈C)(x∈A and y∈B) and (x∈A and y∈C)(x,y) ∈ (A x B) and (x,y) ∈(A x C) // by Cartesian Product. (x,y) ∈ (AxB)∩(AxC)

Prob. Let A,B,C be any three sets, then prove that-AX(B∩C) = (AXB) ∩ (AXC) Solution:(x,y) ∈ Ax(B∩C)x∈A and (y∈(B∩C))x∈A and (y∈B and y∈C)(x∈A and y∈B) and (x∈A and y∈C)(x,y) ∈ (A x B) and (x,y) ∈(A x C) // by Cartesian Product. (x,y) ∈ (AxB)∩(AxC) Prob. Prove that-A∩(B∪C) = (A∩B) ∪ (A∩C) Solution:Let x ∈ […]

Mathematical induction is a unique and special way to prove the things, in only two steps. Step 1. Show that it is true for n = 1.Step 2. Show that if n = k is true then n = k+1 is also true. For example:Prob. By principal of mathematical induction prove that11n+2 + 122n+1 is divisible by 133, n ∈ […]

A set is a collection of definite well defined objects.A set is a collection of objects which are distinct from each other. A set is usually denoted by capital letter, i.e, A, B, S, T, G etc.A set elements are denoted by small letter, i.e, a, b, s, t etc. CONSTRUCTION OF SET: In construction […]