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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.

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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

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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 ∨.

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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

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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.

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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)

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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.

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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.

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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.

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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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