Read the assignment carefully. Write your solution with all steps explained!
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(version: November 1, 2016)
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Exercises for the sixth test
Realtions and their properties.
6.1. Let A be the set of all lines in a plane. On A we define relation R so that: ∀p, q ∈ A : p R q ⇔ p k q.
Shaw what properties relation R has. (Consider coinciding lines as parallel.)
6.2. Let A be the set of all lines in a plane. On A we define relation R so that: ∀p, q ∈ A : p R q ⇔ p is
perpendicular to q.
Shaw what properties relation R has.
6.3. As a set we consider a group of all students in a lecture room S.
Each student is in relation R with himself and two students x, y ∈ S (in this order) are in relation R,
if y sits on the left next to x in the same row.(Viewed from the position of the lecturer.) Shaw, that the
relation R is a partial ordering. What are minimal and maximal elements in S?
6.4. Consider a set Ai with divisibility relation |. Shaw, whether it is (is not) partial ordering if:
(a) A1 = [0, 9] ,
(b) A2 = {−2, 2, 4, 6, 8, 10}.
Divisibility relation b divides a“ (denoted by b | a) if and only if there exists c such that b · c = a, where
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a, b, c ∈ N0 in part (a), and or a, b, c ∈ Z in part (b).
6.5. Sketch Hasse diagram of the inclusion relation on the set 2A \ ∅, where A = {1, 2, 3}. List all maximal,
minimal, the greatest and the smallest elements of this relation.
6.6. Is the relation being kindred“ defined on the set of all living people equivalence relation? (We suppose,
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that any man is kindred with himself.) Justify your decision.
6.7. For any two nonzero vectors (x, y), (a, b) ∈ R2 \ {(0, 0)} we set (x, y) R (a, b) ⇔ there exists a k ∈ R
such that (x, y) = k(a, b).
(We call such a pair of vectors colinear.) Shaw that R is equivalence relation.
6.8. For any two vectors (x, y, z), (a, b, c) ∈ R3 holds (x, y, z) R (a, b, c) ⇔ xa + yb + zc = 0. What properties
relation R has? Justify your answers.
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6.9. Consider the set X = {0, 1}. We define relation on Y = 2X so that: ∀A, B ∈ Y : A R B ⇔ A ⊆ B.
(a) Is the relation R partial ordering relation? Justify your answer!
(b) If yeas, sketch Hasse diagram of the relation.
(c) If possible, find in A with R relation all minimal and minimum, maximal and maximum elaments.