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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.**

- Meet
- Join
- Both A and B
- 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 -**

- Commutative law
- Associative law
- Absorption law
- All of the above

**Answer:** D) All of the above

**Explanation:**

Axioms that lattice L holds are -

- Commutative law
- Associative law
- Absorption law

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

- Mixing
- Interchanging
- Adding
- 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:**

- a ∨ 1 = 1
- a ∧1= a
- a ∨0=a
- All of the above

**Answer:** D) All of the above

**Explanation:**

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

- a ∨ 1 = 1
- a ∧1= a
- a ∨0=a

**5. A non-empty subset L _{1} of a lattice L is considered. It can be realized that L_{1} is a ____-lattice of L if it itself is a lattice, i.e., whenever a ∨ b ∈ L_{1} and a ∧ b ∈ L_{1} whenever a ∈ L_{1} and b ∈ L_{1}.**

- Super
- Sub
- Side
- None

**Answer:** B) Sub

**Explanation:**

A non-empty subset L_{1} of a lattice L is considered. It can be realized that L_{1} is a sub-lattice of L if it itself is a lattice, i.e., whenever a ∨ b ∈ L_{1} and a ∧ b ∈ L_{1} whenever a ∈ L_{1} and b ∈ L_{1}.

**6. Two lattices L _{1} and L_{2} are called isomorphic lattices if there is a ____ from L_{1} to L2.**

- Dijection
- Bijection
- Rejection
- None

**Answer:** B) Bijection

**Explanation:**

Two lattices L_{1} and L_{2} are called isomorphic lattices if there is a bijection from L_{1} to L_{2}.

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

- a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
- a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)
- Both A and B
- 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:

- a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
- a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

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

- Non-distributive
- Distributive
- Associative
- 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 _____.**

- a ∨ x = I
- a ∧ x = 0
- Both A and B
- 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.**

- Complimented lattices
- Uncomplimented lattices
- Conceptional
- Unconceptional lattices

**Answer:** A) Complimented lattices

**Explanation:**

Complimented lattices are bounded and contain complements for each element.

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

- b ≤ c
- a ≤ b
- a ≤ c
- 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 = L _{1} x L_{2} where the binary operations (join) and (meet) on L are such that for any (a_{1},b_{1}) and (a_{2},b_{2}) in L.**

- Direct product
- Indirect product
- Direct sum
- Indirect sum

**Answer:** A) Direct product

**Explanation:**

(L, ∧,∨) is the direct product of lattices, where L = L_{1} x L_{2} where the binary operations (join) and (meet) on L are such that for any (a_{1},b_{1}) and (a_{2},b_{2}) in L.

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