Virtual Labs

A relation R on set X is reflexive if

a: (a, a) in R ∀ a in X Explanation

Explanation

b: aRb implies bRa, ∀ a, b in X Explanation

Explanation

c: (a, b) in R ∀ a,b in X Explanation

Explanation

d: If (a, b) in R and (b, c) in R then, (a, c) in R, ∀ a, b, c in X Explanation

Explanation

A relation R on set X is transitive if

a: (a, a) in R ∀ a in X Explanation

Explanation

b: aRb implies bRa, ∀ a, b in X Explanation

Explanation

c: (a, b) in R ∀ a,b in X Explanation

Explanation

d: If (a, b) in R and (b, c) in R then, (a, c) in R, ∀ a, b, c in X Explanation

Explanation

A relation R is symmetric if

a: (a, a) in R ∀ a in X Explanation

Explanation

b: aRb implies bRa, ∀ a, b in X Explanation

Explanation

c: (a, b) in R ∀ a,b in X Explanation

Explanation

d: If (a, b) in R and (b, c) in R then, (a, c) in R, ∀ a, b, c in X Explanation

Explanation

In a Hasse diagram of a partially ordered set, which edges can be omitted without losing information about the ordering?

a: Edges representing reflexive relations (loops) Explanation

Explanation

b: Edges that can be inferred through transitivity Explanation

Explanation

c: Edges between incomparable elements Explanation

Explanation

d: Edges representing symmetric relations Explanation

Explanation

Given a partially ordered set (X, ≤), an element m ∈ X is called maximal if:

a: For all x ∈ X, x ≤ m Explanation

Explanation

b: For all x ∈ X, m ≤ x Explanation

Explanation

c: For all x ∈ X, if m ≤ x then x = m Explanation

Explanation

d: For all x ∈ X, if x ≤ m then x = m Explanation

Explanation

Which property distinguishes a partial order from an equivalence relation?

a: Reflexivity Explanation

Explanation

b: Transitivity Explanation

Explanation

c: Antisymmetry instead of symmetry Explanation

Explanation

d: None of the above Explanation

Explanation

In a lattice (L, ≤), what is true about the meet (∧) and join (∨) of any two elements a, b ∈ L?

a: a ∧ b is always less than both a and b; a ∨ b is always greater than both a and b Explanation

Explanation

b: a ∧ b is the unique greatest lower bound; a ∨ b is the unique least upper bound Explanation

Explanation

c: a ∧ b = b ∧ a and a ∨ b = b ∨ a Explanation

Explanation

d: All of the above Explanation

Explanation

Given a Hasse diagram, what can we conclude if there exists no path between elements a and b?

a: a and b must be equal elements Explanation

Explanation

b: a and b are incomparable elements Explanation

Explanation

c: Either a < b or b < a must be true Explanation

Explanation

d: a and b cannot be in the same partially ordered set Explanation

Explanation

What is the relationship between a chain and an antichain in a partially ordered set?

a: Every element must belong to either a chain or an antichain Explanation

Explanation

b: The intersection of a chain and an antichain can contain at most one element Explanation

Explanation

c: The union of all maximal chains equals the union of all maximal antichains Explanation

Explanation

d: Chains and antichains must have the same size in any finite partial order Explanation

Explanation

Parial Order and Hasse Diagram