Question

let A={2,4,12,20} consider the
partial order relation 'divisibility' on A the greatest lower bound for 12 and 20 is​

Answer 1

Step-by-step explanation:

let A={2,4,12,20} consider the

partial order relation 'divisibility' on A the greatest lower bound for 12 and 20 is

Answer 2

Answer:

gcd(12,20) = 4

Step-by-step explanation:

What is partial order relation?

• A relation R is said to be a partial order set or a POSET, if it is,

1. Reflexive, i.e., ∀a∈A , (a,a)∈R
2. Anti Symmetric, i.e., (a,b)∈R, (b,a)∈R ⇒ a=b ∀a,b∈A
3. Transitive, i.e., (a,b)∈R, (b,c)∈R ⇒ (a,c)∈R ∀a,b,c∈A

• It is usually denoted by (P,≤)

Given, A={2,4,12,20}

(2,4)∈R

∵ 2 is divisible by 4 (4 divides 2)

Similarly, finding other divisibility relations of A,

(P,≤) = {(2,4),(2,12),(2,20),(4,12),(4,20),(12,20)}

Next, we find the greatest lower bound of 12 and 20,

To find greatest lower bound of two numbers, we take the gcd of them,

i.e.,  greatest lower bound of 12 and 20 = gcd{12,20}

• GCD - greatest common divisor is the greatest number that can divide two numbers evenly. That is, GCD of 24 and 36 is

                                     $24 = 2*2*2*3\\\\36=2*2*3*3$

                        GCD = multiplication of common factors

                                             $= 2*2*3=12$

∴ gcd(12,20) = 4

(#SPJ3)