1)Consider the equivalence relation {(x,y)|x≡y(mod5)} on the set {n∈Z|−10⩽n⩽5}. Use the roster method to describe the following equivalence classes. [−10]= [−9]=
1)Consider the equivalence relation
on the set {n∈Z|−10⩽n⩽5}.
Use the roster method to describe the following equivalence classes.
[−10]=
[−9]=
2)The partially ordered set (Z+,≼) is the set of positive integers with the following partial order:
a≼b if and only if a=b or a∣b (i.e., a divides b).
Using the partial ordering ≼ on Z+, we find that:
A. 56≺86
B. 56≻86
C. 56 and 86 are incomparable
D. None of the above
Using the partial ordering ≼ on Z+, we find that:
A. 72≻12
B. 72≺12
C. 72 and 12 are incomparable
D. None of the above
Using the partial ordering ≼ on Z+, we find that:
A. 35≻47
B. 35≺47
C. 35 and 47 are incomparable
D. None of the above
Using the partial ordering ≼ on Z+, we find that:
A. 17≻51
B. 17≺51
C. 17 and 51 are incomparable
D. None of the above
3) Consider the set A={1,4,7,8}. The partially ordered set (P(A),⊆) is the power set of A with elements partially ordered by inclusion.
Which of the following sets are less than or equal to {4,8} in the inclusion order on P(A)?
Select all that apply.
A. {1}
B. {1,4}
C. {4,7}
D. {1,7}
E. {8}
F. {7}
G. {4,8}
H. {1,7,8}
I. {1,4,7,8}
J. ∅
K. {1,4,7}
L. {4}
M. {1,8}
N. {7,8}
O. {4,7,8}
P. {1,4,8}
Which of the following sets are incomparable with {4,8} in the inclusion order on P(A)?
Select all that apply.
A. ∅
B. {8}
C. {4,7}
D. {1,4,8}
E. {1,4,7,8}
F. {4}
G. {7}
H. {7,8}
I. {1,7,8}
J. {4,7,8}
K. {1,7}
L. {4,8}
M. {1}
N. {1,4,7}
O. {1,8}
P. {1,4}
4) Let A={1,7,8,9} be a partially ordered set with the usual ⩽ order relation.
Consider the partially ordered set (A×A,≼), where A×A is the Cartesian product of the set A with itself and ≼ is the lexicographic ordering constructed from the usual order relation ⩽.
Select all the elements of A×A that are greater than or equal to (1,9) and smaller than or equal to (8,7) in the ordering ≼.
A. (7,8)
B. (7,1)
C. (9,8)
D. (8,8)
E. (9,1)
F. (1,9)
G. (7,9)
H. (1,1)
I. (1,8)
J. (9,7)
K. (7,7)
L. (8,1)
M. (1,7)
N. (8,9)
O. (9,9)
P. (8,7)
5) Which of the following relations on the set {0,1,2,4,7,8} are antisymmetric? Select all that apply.
A. {(7,4),(0,0),(2,7),(0,4),(1,1),(1,8),(0,1),(1,2),(2,8),(8,4)}
B. {(2,1),(2,2),(0,1),(1,4),(7,8),(4,1),(1,2),(0,2),(8,4),(0,7)}
C. {(7,0),(4,2),(0,1),(7,8),(8,4),(1,1),(8,1),(7,1),(2,1)}
D. {(1,1),(4,7),(7,1),(4,1),(2,1),(0,0),(0,2),(2,0),(7,8),(0,7),(4,0)}
E. {(4,7),(2,4),(8,7),(2,7),(0,2),(2,1),(8,4),(1,7),(0,7),(8,0)}
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