{"id":2920,"date":"2025-06-06T14:47:54","date_gmt":"2025-06-06T14:47:54","guid":{"rendered":"https:\/\/diznr.com\/?p=2920"},"modified":"2025-06-06T14:47:54","modified_gmt":"2025-06-06T14:47:54","slug":"day-06part-04-discrete-mathematics-groupoid-semi-group-monoid-group-and-group-abelian","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/day-06part-04-discrete-mathematics-groupoid-semi-group-monoid-group-and-group-abelian\/","title":{"rendered":"Day 06Part 04 – Discrete Mathematics – Groupoid , Semi group , monoid , Group and abelian group"},"content":{"rendered":"

Day 06Part 04 – Discrete Mathematics – Groupoid , Semi group , monoid , Group and abelian group<\/p>\n

[fvplayer id=”173″]<\/p>\n

\u00a0Discrete Mathematics \u2013 Groupoid, Semigroup, Monoid, Group & Abelian Group<\/strong><\/h3>\n

In Discrete Mathematics & Abstract Algebra<\/strong>, different algebraic structures help in understanding mathematical operations and their properties. Let\u2019s go step by step!<\/p>\n

\u00a0Groupoid<\/strong><\/h3>\n

A groupoid<\/strong> is a non-empty set<\/strong> with a binary operation<\/strong> (a rule that combines two elements to form another element in the set).<\/p>\n

Definition:<\/strong> A set G<\/strong> with a binary operation (*)<\/strong> is called a groupoid<\/strong> if:<\/p>\n

\u2200a,b\u2208G,a\u2217b\u00a0is\u00a0also\u00a0in\u00a0G.\\forall a, b \\in G, \\quad a * b \\text{ is also in } G.<\/span>\u2200<\/span>a<\/span>,<\/span>b<\/span>\u2208<\/span><\/span>G<\/span>,<\/span>a<\/span>\u2217<\/span><\/span>b<\/span>\u00a0is\u00a0also\u00a0in\u00a0<\/span><\/span>G<\/span>.<\/span><\/span><\/span><\/span><\/span><\/p>\n

Example:<\/strong> The set of natural numbers (N, +)<\/strong> is a groupoid since the sum of any two natural numbers is also a natural number.<\/p>\n

\u00a0Semigroup<\/strong><\/h3>\n

A semigroup<\/strong> is a groupoid<\/strong> with the associative property<\/strong>.<\/p>\n

Definition:<\/strong> A set S<\/strong> with a binary operation (*)<\/strong> is a semigroup<\/strong> if:<\/p>\n

\u2200a,b,c\u2208S,(a\u2217b)\u2217c=a\u2217(b\u2217c).\\forall a, b, c \\in S, \\quad (a * b) * c = a * (b * c).<\/span>\u2200<\/span>a<\/span>,<\/span>b<\/span>,<\/span>c<\/span>\u2208<\/span><\/span>S<\/span>,<\/span>(<\/span>a<\/span>\u2217<\/span><\/span>b<\/span>)<\/span>\u2217<\/span><\/span>c<\/span>=<\/span><\/span>a<\/span>\u2217<\/span><\/span>(<\/span>b<\/span>\u2217<\/span><\/span>c<\/span>)<\/span>.<\/span><\/span><\/span><\/span><\/span><\/p>\n

Example:<\/strong> The set of natural numbers (N, +)<\/strong> is a semigroup because addition is associative:<\/p>\n

(2+3)+4=2+(3+4).(2 + 3) + 4 = 2 + (3 + 4).<\/span>(<\/span>2<\/span>+<\/span><\/span>3<\/span>)<\/span>+<\/span><\/span>4<\/span>=<\/span><\/span>2<\/span>+<\/span><\/span>(<\/span>3<\/span>+<\/span><\/span>4<\/span>)<\/span>.<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u00a0Monoid<\/strong><\/h3>\n

A monoid<\/strong> is a semigroup<\/strong> with an identity element<\/strong>.<\/p>\n

Definition:<\/strong> A set M<\/strong> with a binary operation (*)<\/strong> is a monoid<\/strong> if:<\/p>\n

    \n
  1. Associativity:<\/strong> (a\u2217b)\u2217c=a\u2217(b\u2217c)(a * b) * c = a * (b * c)<\/span>(<\/span>a<\/span>\u2217<\/span><\/span>b<\/span>)<\/span>\u2217<\/span><\/span>c<\/span>=<\/span><\/span>a<\/span>\u2217<\/span><\/span>(<\/span>b<\/span>\u2217<\/span><\/span>c<\/span>)<\/span><\/span><\/span><\/span> for all a,b,c\u2208Ma, b, c \\in M<\/span>a<\/span>,<\/span>b<\/span>,<\/span>c<\/span>\u2208<\/span><\/span>M<\/span><\/span><\/span><\/span>.<\/li>\n
  2. Existence of an Identity Element (e):<\/strong> There exists an element e\u2208Me \\in M<\/span>e<\/span>\u2208<\/span><\/span>M<\/span><\/span><\/span><\/span> such that a\u2217e=e\u2217a=a,\u2200a\u2208M.a * e = e * a = a, \\quad \\forall a \\in M.<\/span>a<\/span>\u2217<\/span><\/span>e<\/span>=<\/span><\/span>e<\/span>\u2217<\/span><\/span>a<\/span>=<\/span><\/span>a<\/span>,<\/span>\u2200<\/span>a<\/span>\u2208<\/span><\/span>M<\/span>.<\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ol>\n

    Example:<\/strong> The set of natural numbers (N, \u00d7)<\/strong> is a monoid because multiplication is associative and has an identity element 1<\/strong>.<\/p>\n

    2\u00d7(3\u00d74)=(2\u00d73)\u00d74,2\u00d71=2.2 \\times (3 \\times 4) = (2 \\times 3) \\times 4, \\quad 2 \\times 1 = 2.<\/span>2<\/span>\u00d7<\/span><\/span>(<\/span>3<\/span>\u00d7<\/span><\/span>4<\/span>)<\/span>=<\/span><\/span>(<\/span>2<\/span>\u00d7<\/span><\/span>3<\/span>)<\/span>\u00d7<\/span><\/span>4<\/span>,<\/span>2<\/span>\u00d7<\/span><\/span>1<\/span>=<\/span><\/span>2.<\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u00a0Group<\/strong><\/h3>\n

    A group<\/strong> is a monoid<\/strong> in which every element has an inverse<\/strong>.<\/p>\n

    Definition:<\/strong> A set G<\/strong> with a binary operation (*)<\/strong> is a group<\/strong> if:<\/p>\n

      \n
    1. Associativity<\/strong>: (a\u2217b)\u2217c=a\u2217(b\u2217c)(a * b) * c = a * (b * c)<\/span>(<\/span>a<\/span>\u2217<\/span><\/span>b<\/span>)<\/span>\u2217<\/span><\/span>c<\/span>=<\/span><\/span>a<\/span>\u2217<\/span><\/span>(<\/span>b<\/span>\u2217<\/span><\/span>c<\/span>)<\/span><\/span><\/span><\/span>, for all a,b,c\u2208Ga, b, c \\in G<\/span>a<\/span>,<\/span>b<\/span>,<\/span>c<\/span>\u2208<\/span><\/span>G<\/span><\/span><\/span><\/span>.<\/li>\n
    2. Identity Element (e):<\/strong> There exists an element e\u2208Ge \\in G<\/span>e<\/span>\u2208<\/span><\/span>G<\/span><\/span><\/span><\/span> such that a\u2217e=e\u2217a=a,\u2200a\u2208G.a * e = e * a = a, \\quad \\forall a \\in G.<\/span>a<\/span>\u2217<\/span><\/span>e<\/span>=<\/span><\/span>e<\/span>\u2217<\/span><\/span>a<\/span>=<\/span><\/span>a<\/span>,<\/span>\u2200<\/span>a<\/span>\u2208<\/span><\/span>G<\/span>.<\/span><\/span><\/span><\/span><\/span><\/li>\n
    3. Inverse Element:<\/strong> For every a\u2208Ga \\in G<\/span>a<\/span>\u2208<\/span><\/span>G<\/span><\/span><\/span><\/span>, there exists an element a\u22121a^{-1}<\/span>a<\/span>\u22121<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> such that a\u2217a\u22121=a\u22121\u2217a=e.a * a^{-1} = a^{-1} * a = e.<\/span>a<\/span>\u2217<\/span><\/span>a<\/span>\u22121<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>a<\/span>\u22121<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2217<\/span><\/span>a<\/span>=<\/span><\/span>e<\/span>.<\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ol>\n

      Example:<\/strong> The set of integers (Z, +)<\/strong> forms a group because:<\/p>\n