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In Discrete Structures, understanding the concepts of sets, relations, and lattices is crucial. Key topics include the definitions and properties of subsets, power sets, and Hasse diagrams. Additionally, knowing how to c…
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Define a Lattice. Draw the Hasse diagram of (D₃₆, /) and Show that it is a lattice.
A lattice is a partially ordered set where any two elements have a unique supremum and an infimum.
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Let R = {(1, 2), (2, 3), (3, 1)} and A = {1, 2, 3} find the reflexive, symmetric and transitive closure of R using composition of relation R.
The reflexive closure adds (1,1), (2,2), (3,3); symmetric closure adds (2,1), (3,2), (1,3); transitive closure adds (1,3), (2,2), (3,1).
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Use Warshall's algorithm to find the transitive closure of the following relation on the set A = {1, 2, 3}. R = {(1, 2), (2, 3), (3, 1)}
Warshall's algorithm iteratively adds paths to find the transitive closure.
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Identify whether the each of the following relation defined on the set A = {1, 2, 3} are reflexive relation, symmetric relation, anti-symmetric relation and transitive relation.
Analyze each relation for reflexivity, symmetry, anti-symmetry, and transitivity.
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Let X = {1, 2, 3, 4, 5, 6, 7} and R= {(x, y)|(x-y) is divisible by 3}, then prove that R is an equivalence relation.
R is reflexive, symmetric, and transitive, hence an equivalence relation.
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If R and S are equivalence relation on the set A, prove that R ∩ S is an equivalence relation.
The intersection of two equivalence relations is also reflexive, symmetric, and transitive.
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If A = {1, 2, 3}, B = {4, 5} and C = {1, 2, 3, 4, 5}. Find (i) A × B (ii) C × B (iii) B × B Hence prove that: (C × B) – (A × B) = B × B.
Calculate Cartesian products and verify the equality.
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Differentiate between proper subset and superset.
A proper subset is a subset that is not equal to the original set, while a superset contains all elements of the subset.
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Define Power set. Let A = {Ø, b} construct the following set: (i) A - Ø (ii) A ∩ P(A).
The power set of A is the set of all subsets of A.
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Let P and Q be the multisets where P= {5.a, 7.b, 8.c} and Q= {3.a, 5.b, 9.c}. Find: (i) P ∪ Q (ii) Q - P.
P ∪ Q = {5.a, 7.b, 8.c, 3.a, 5.b, 9.c}; Q - P = {3.a, 5.b, 9.c}.
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Define Greatest element and Least Element in Hasse diagram.
Greatest element is the unique element greater than all others; least element is the unique element less than all others.
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Let A = {1, 2, 3, 4} and B = {a, b, c, d, e}. Compute the total number of possible relations from A to B.
The total number of possible relations from A to B is $2^{|A| imes |B|} = 2^{4 imes 5} = 2^{20} = 1048576$.
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