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Questions and Answers
In a POSET, what is the greatest lower bound element for vertices 4 and 5?
What is the least upper bound element for the pair of vertices {4, 5} in a join semilattice?
Which element is the greatest lower bound element for the pair {c, d} in a meet semilattice?
What type of lattice is defined as both a join semilattice and a meet semilattice?
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Which vertex is NOT a lower bound of vertices 4 and 5?
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In a POSET, what is the least upper bound element for vertices {3, 4}?
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Which of the following pairs has vertex a as the greatest lower bound element in a meet semilattice?
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What makes a POSET a join semilattice?
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What mathematical concept is visually represented by Hasse diagrams?
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Which of the following properties defines a function in calculus?
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What type of lattice is defined as both a join semilattice and a meet semilattice?
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What makes a POSET a join semilattice?
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In the context of POSETS, what is the greatest lower bound element?
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In a gas station analogy, what represents the input values (x) and output values (y) in a function?
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What function property ensures that for a specific input, we always get the same output?
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Which of the following is an example of a function in calculus?
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What is the relationship between lattices and semilattices in the context of POSETS?
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In the context of POSETS, what is the least upper bound element?
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Study Notes
Hasse Diagram
- A Hasse diagram is a graphical representation of a Partially Ordered Set (POSET).
- It shows elements as vertices and relations as edges, omitting self-loops and transitive edges to simplify the graph.
- Vertices with no incoming edges are called maximal elements, and vertices with no outgoing edges are called minimal elements.
- A directed edge represents the direction of the relationship, and the levels of elements in the diagram indicate the hierarchy of relations.
Partially Ordered Sets (POSETS)
- A relation R over a set A is a POSET if it is reflexive, anti-symmetric, and transitive.
- Reflexive relation: every element in A is related to itself.
- Anti-symmetric relation: for any elements a and b in A, if (a, b) belongs to R and (b, a) belongs to R, then a must equal b.
- Transitive relation: for any elements a, b, and c in A, if (a, b) and (b, c) belong to R, then (a, c) must also belong to R.
Semilattices and Lattices
- A POSET is a join semilattice if every pair of elements has a least upper bound element.
- A POSET is a meet semilattice if every pair of elements has a greatest lower bound element.
- A POSET is a lattice if it is both a join semilattice and meet semilattice.
Set Theory Terminology
- Set: an unordered collection of objects.
- Ordered Pair: a pair of numbers (x, y) written in a specific order, distinct from (y, x).
- Cartesian Product: the product of two sets A and B, forming ordered pairs of every element from A with every element from B.
Basics of a Function
- Function definition: a mathematical relationship between two sets of numbers, where each input value (x) is associated with a unique output value (y).
- Function example: a "black box" that takes an input (x) and produces an output (y).
- Gas station analogy: functions are encountered in daily life, where the output depends on the input.
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Description
Learn how to represent a POSET using a Hasse diagram. Perform Cartesian products of sets and create relations. Understand the process step by step with a given example.
