Discrete Mathematics | Lattices MCQs

Discrete Mathematics | Lattices MCQs: This section contains multiple-choice questions and answers on Lattices in Discrete Mathematics.
Submitted by Anushree Goswami, on November 03, 2022

1. Assume that L is a non-empty set closed only under two binary operations, ____, denoted by ∧ and ∨. It is called a lattice if L has a, b, and c elements, where a, b, and c are the elements in L.

  1. Meet
  2. Join
  3. Both A and B
  4. None of the above

Answer: C) Both A and B

Explanation:

Assume that L is a non-empty set closed only under two binary operations, meet and join, denoted by ∧ and ∨. It is called a lattice if L has a, b, and c elements, where a, b, and c are the elements in L.


2. Axioms that lattice L holds are -

  1. Commutative law
  2. Associative law
  3. Absorption law
  4. All of the above

Answer: D) All of the above

Explanation:

Axioms that lattice L holds are -

  1. Commutative law
  2. Associative law
  3. Absorption law

3. An expression's dual is the expression that can be obtained by _____ ∧ and ∨.

  1. Mixing
  2. Interchanging
  3. Adding
  4. Removing

Answer: B) Interchanging

Explanation:

An expression's dual is the expression that can be obtained by interchanging ∧ and ∨.


4. The following identities are valid if L is a bounded lattice:

  1. a ∨ 1 = 1
  2. a ∧1= a
  3. a ∨0=a
  4. All of the above

Answer: D) All of the above

Explanation:

The following identities are valid if L is a bounded lattice:

  1. a ∨ 1 = 1
  2. a ∧1= a
  3. a ∨0=a

5. A non-empty subset L1 of a lattice L is considered. It can be realized that L1 is a ____-lattice of L if it itself is a lattice, i.e., whenever a ∨ b ∈ L1 and a ∧ b ∈ L1 whenever a ∈ L1 and b ∈ L1.

  1. Super
  2. Sub
  3. Side
  4. None

Answer: B) Sub

Explanation:

A non-empty subset L1 of a lattice L is considered. It can be realized that L1 is a sub-lattice of L if it itself is a lattice, i.e., whenever a ∨ b ∈ L1 and a ∧ b ∈ L1 whenever a ∈ L1 and b ∈ L1.


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6. Two lattices L1 and L2 are called isomorphic lattices if there is a ____ from L1 to L2.

  1. Dijection
  2. Bijection
  3. Rejection
  4. None

Answer: B) Bijection

Explanation:

Two lattices L1 and L2 are called isomorphic lattices if there is a bijection from L1 to L2.


7. If any element a, b, or c of a lattice L satisfies the following distributive properties, it is called a distributive lattice:

  1. a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
  2. a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)
  3. Both A and B
  4. None of the above

Answer: C) Both A and B

Explanation:

If any element a, b, or c of a lattice L satisfies the following distributive properties, it is called a distributive lattice:

  1. a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
  2. a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

8. ____ lattices are those that do not satisfy the above properties.

  1. Non-distributive
  2. Distributive
  3. Associative
  4. Non-associative

Answer: A) Non-distributive

Explanation:

Non-distributive lattices are those that do not satisfy the above properties.


9. An upper bound I and a lower bound o define a bounded lattice L. Assume that a is an element if L. In L, a complementary element x is an element within an if _____.

  1. a ∨ x = I
  2. a ∧ x = 0
  3. Both A and B
  4. None of the above

Answer: C) Both A and B

Explanation:

An upper bound I and a lower bound o define a bounded lattice L. Assume that a is an element if L. In L, a complementary element x is an element within an if a ∨ x = I and a ∧ x = 0.


10. _____ are bounded and contain complements for each element.

  1. Complimented lattices
  2. Uncomplimented lattices
  3. Conceptional
  4. Unconceptional lattices

Answer: A) Complimented lattices

Explanation:

Complimented lattices are bounded and contain complements for each element.


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11. (L, b, c) is a modular lattice if a ∨ (b ∧ c) = (a ∨ b) ∧ c whenever ____.

  1. b ≤ c
  2. a ≤ b
  3. a ≤ c
  4. None

Answer: C) a ≤ c

Explanation:

(L, b, c) is a modular lattice if a ∨ (b ∧ c) = (a ∨ b) ∧ c whenever a ≤ c.


12. (L, ∧,∨) is the ____ of lattices, where L = L1 x L2 where the binary operations (join) and (meet) on L are such that for any (a1,b1) and (a2,b2) in L.

  1. Direct product
  2. Indirect product
  3. Direct sum
  4. Indirect sum

Answer: A) Direct product

Explanation:

(L, ∧,∨) is the direct product of lattices, where L = L1 x L2 where the binary operations (join) and (meet) on L are such that for any (a1,b1) and (a2,b2) in L.



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