UGC NET CSE | December 2019 | Part 2 | Question: 4

Question

What are the greatest lower bound $\text{(GLB)}$ and the least upper bound $\text{(LUB)}$ of the sets $A= \{ 3, 9, 12 \}$ and $B=\{1,2,4,5,10 \}$ if they exist in poset $(z^+, / )$?

• A. $\text{A(GLB – 3, LUB – 36); B(GLB – 1, LUB – 20)}$
• B. $\text{A(GLB – 3, LUB – 12); B(GLB – 1, LUB – 10)}$
• C. $\text{A(GLB – 1, LUB – 36); B(GLB – 2, LUB – 20)}$
• D. $\text{A(GLB – 1, LUB – 12); B(GLB – 2, LUB – 10)}$

Answer 1

• A. A(GLB-3, LUB – 36); B(GLB-1, LUB – 20)

What is poset(z+. /)?

Partially ordered set (poset) of all positive integers with the relation “/” which means “divides”.
z+ is set of positive integers. So this set contains {1,2,3,4,5,6,7,…..}
E.g. 6/3 = 2, thus 3 divides 6

What is Least Upper Bound for given set?

Find a value u such that for every element s of set S, s <relation> u exists.
For {3,9,12} – Find least (smallest) number (say M), for which 3, 9, 12 “divides” M. Because our relation is “/”, this is equivalent to finding LCM (Least Common Multiple) of 3, 9, 12. Thus we get LUB as 12x3=9x4=36. (3, 9 and 12 divide 36)

What is Greatest Lower Bound?

Find value l such that for every element s of set S, l <relation> s exists
For {3,9,12} – Find greatest number less than or equal to an element within {3,9,12}, which “divides” {3,9,12}
Thus GLB is 3 as 3 “divides” {3,9,12}. Because our relation is “/”, this is equivalent to finding HCF (Highest Common Factor) of 3, 9, 12.

[NOTE: GLB and LUB are values which must belong to the poset (z+,/) but need not necessarily belong to the set which they bound.]

Explanation of poset on Wolfram MathWorld 
Definition of poset on Wolfram MathWorld
[Here, the symbol “ ≤ “ refers to the relation e.g. for our case, the relation is “/” or “divides”.]

Answer 2

OPTION 1: IS CORRECT

GLB(greatest lower bound )

all LB’S X<= X €A <= All elements of B

LUB(least upper bound )

All elements € B<= X€A<= all UB’S of B

That means we can take glb=gcd

and lub=lcm

so simply find the lcm  for lub and gcd for glb

in this question

A={3,9 ,12}

SO gcd=3

and lcm=36

B={1,2,4,5,10}

gcd = glb=1

lcm = lub=20

so option 1 is correct