Author: admin
-
Boolean Expression
Consider a Boolean algebra (B, ∨,∧,’,0,1).A Boolean expression over Boolean algebra B is defined as Example: Consider a Boolean algebra ({0, 1, 2, 3},∨,∧,’,0,1). are Boolean expressions over the Boolean Algebra. A Boolean expression that contains n distinct variables is usually referred to as a Boolean expression of n variables.Backward Skip 10sPlay VideoForward Skip…
-
Boolean Algebra:
A complemented distributive lattice is known as a Boolean Algebra. It is denoted by (B, ∧,∨,’,0,1), where B is a set on which two binary operations ∧ (*) and ∨(+) and a unary operation (complement) are defined. Here 0 and 1 are two distinct elements of B. Since (B,∧,∨) is a complemented distributive lattice, therefore…
-
Lattices
Let L be a non-empty set closed under two binary operations called meet and join, denoted by ∧ and ∨. Then L is called a lattice if the following axioms hold where a, b, c are elements in L: 1) Commutative Law: –(a) a ∧ b = b ∧ a …
-
Hasse Diagrams
It is a useful tool, which completely describes the associated partial order. Therefore, it is also called an ordering diagram. It is very easy to convert a directed graph of a relation on a set A to an equivalent Hasse diagram. Therefore, while drawing a Hasse diagram following points must be remembered. The Hasse diagram…
-
Partially Ordered Sets
Consider a relation R on a set S satisfying the following properties: Then R is called a partial order relation, and the set S together with partial order is called a partially order set or POSET and is denoted by (S, ≤). Example: Elements of POSET: Note: There can be more than one maximal or…
-
Normal SubGroup
Let G be a group. A subgroup H of G is said to be a normal subgroup of G if for all h∈ H and x∈ G, x h x-1∈ H If x H x-1 = {x h x-1| h ∈ H} then H is normal in G if and only if xH x-1⊆H, ∀ x∈…
-
Subgroup
If a non-void subset H of a group G is itself a group under the operation of G, we say H is a subgroup of G. Theorem: – A subset H of a group G is a subgroup of G if: Cyclic Subgroup:- A Subgroup K of a group G is said to be cyclic subgroup…
-
Group
Let G be a non-void set with a binary operation * that assigns to each ordered pair (a, b) of elements of G an element of G denoted by a * b. We say that G is a group under the binary operation * if the following three properties are satisfied: 1) Associativity: The binary operation…
-
SemiGroup
Let us consider, an algebraic system (A, *), where * is a binary operation on A. Then, the system (A, *) is said to be semi-group if it satisfies the following properties: Example: Consider an algebraic system (A, *), where A = {1, 3, 5, 7, 9….}, the set of positive odd integers and * is…
-
Properties of Binary Operations
There are many properties of the binary operations which are as follows: 1. Closure Property: Consider a non-empty set A and a binary operation * on A. Then is closed under the operation *, if a * b ∈ A, where a and b are elements of A. Example1: The operation of addition on the set of…