Partial Orders MCQ Quiz in বাংলা - Objective Question with Answer for Partial Orders - বিনামূল্যে ডাউনলোড করুন [PDF]

Concept:

A partial order set, or poset, is a chain if for every a and b in the set, either a ≤ b or b ≤ a.

Explanation:

(3) is correct

Key Points

 The given lattice is,

I ) The upper bound is B

False, there is no upper bound of the above  lattice

II) minimal element is O

True, the Minimal element of the above lattice is 0 only. The meet of above lattice at only o.

III) maximal element s are A,B,C

True, The maximal elements of the above lattice is  A, B, C.

IV) Minimal elements are O, N

False, the Minimal element of the above lattice is 0 only. The meet of above lattice at only o.

Hence the correct answer is II, III are correct.

Given lattice is Boolean Algebra

Number of vertices = 2ⁿ = 2³ = 8

Number of edges = n × 2^{n - 1} = 3 × 2^3-1 = 12

Diagram:

Vertices = 8

Edges = 12

Let X = {1,2,3} then POSET POSET [ {1,2,3}; ≤]

Hasse Diagram

Let A = {x, y} then P(A) = {ϕ , {x} , {y} , {x , y}}

Hasse diagram:

It is both meet semi lattice and join semi lattice ∴ it is a lattice

Note:

Although we cannot generalize with just only one example but both the POSET are lattice.

The Hasse diagram for the given poset is:

Therefore, maximal elements are {12, 20, 25} and the minimal elements are {2, 5}.

LB (a, b) = {d, g, h, 0, i}

LB (b, c) = {g, h, i, 0, f}

∴ \(L\cap R\) = {g, h, i, 0}

→ LUB of (9, 1) = 9

∴ 1 cannot be its complement

→ LUB of (9, 2) = 36

GLB of (9, 2) = 1

∴ 2 is its complement

→ LUB of (9, 3) = 9

∴ 3 cannot be its complement

→ LUB of (9, 4) = 36

GLB of (9, 4) = 1

∴ 4 is its complement

→ LUB of (9, 6) = 36

GUB of (9, 6) = 3

∴ 6 cannot be its complement

→ LUB of (9, 36) = 36

GUB of (9, 36) = 9

∴ 36 cannot be its complement

Complement 9 are: 2 and 4

Important Points:

GLB is greatest lower bound

Given:

A = {x, y}

Calculation:

Subsets of A are ϕ, {x}, {y}, {x, y}

 ∴ P(A) = {ϕ, {x}, {y} {x, y}}

Correct answer is option 4.

Hasse diagram of given Lattice:

Option 1. {1, 2, 3, 6}

LUB of (1, 2) = 2, GLB of (1, 2) = 1

LUB of (1, 3) = 3, GLB of (1, 3) = 1

LUB of (1, 6) = 6, GLB of (1, 6) = 1

LUB of (2, 3) = 6, GLB of (2, 3) = 1

LUB of (3, 6) = 6, GLB of (3, 6) = 3

Since LUB and GLB of each pair is same as lattice L

Therefore, it is a sub lattice.

Option 2: {1, 2, 6, 9}

It is not lattice because 6 and 9 are two maximal elements in the POSET {1, 2, 6, 9}

If it is not lattice therefore it cannot be sublattice

Option 3:  { 1, 4, 9, 36 }

LUB of (1, 4) = 4, GLB of (1, 4) = 1

LUB of (1, 9) = 9, GLB of (1, 9) = 1

LUB of (1, 36) = 36, GLB of (1, 36) = 1

LUB of (4, 9) = 36, GLB of (4, 9) = 1

LUB of (4, 36) = 36, GLB of (4, 36) = 4

LUB of (9, 36) = 36, GLB of (9, 36) = 9

Since LUB and GLB of each pair is same as lattice L. Therefore, it is a sublattice.

Option 4: {1, 2, 4, 6}

It is not lattice because 4 and 6 are two maximal elements in the POSET {1, 2, 4, 6}

If it is not lattice therefore it cannot be sublattice

Hence only 1 and 3 are sub lattice.

Important Points:

GLB is greatest lower bound

LUB is least upper bound

Partial ordering relation is

• Reflexive
• Anti-symmetric
• Transitive

It is not asymmetric.