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There are 3 parts to this question. The answer choices for each are all. 1) reflexive 2) irreflexive 3) symmetric 4) anti-symmetric 5) asymmetric 6) transitive 7) connected Part A) consider the relation L in example 3.2. Choose all the properties L has. Part B) let I be the identity relation on N in example 3.2. Choose all properties which I has.

There are 3 parts to this question. The answer choices for each are all. 1) reflexive 2) irreflexive 3) symmetric 4) anti-symmetric 5) asymmetric 6) transitive 7) connected Part A) consider the relation L in example 3.2. Choose all the properties L has. Part B) let I be the identity relation on N in example 3.2. Choose all properties which I has.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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There are 3 parts to this question. The answer choices for each are all. 1) reflexive 2) irreflexive 3) symmetric 4) anti-symmetric 5) asymmetric 6) transitive 7) connected Part A) consider the relation L in example 3.2. Choose all the properties L has. Part B) let I be the identity relation on N in example 3.2. Choose all properties which I has. Part C) consider the relation K in example 3.2 defined by K=LUI where I is the identity relation. Choose all properties K has.
Example 3.2. The set N² of pairs of natural numbers can be listed in a 2-dimensional matrix like this: (0,0) (0,1) (0,2) (0,3) (1,0) (1,1) (1,2) (1,3) (2,0) (3,0) (3,1) (3,2) (3,3) (2,1) (2,2) (2,3) ... ... We have put the diagonal, here, in bold, since the subset of N2 consisting of the pairs lying on the diagonal, i.e., {{0,0), (1,1), (2, 2),...}, is the identity relation on N. (Since the identity relation is popular, let's define Id, {{x,x) : x € X} for any set A.) The subset of all pairs lying above the diagonal, i.e., L = {(0,1), (0,2),..., (1,2), (1,3),..., (2, 3), (2,4), .},
Transcribed Image Text:Example 3.2. The set N² of pairs of natural numbers can be listed in a 2-dimensional matrix like this: (0,0) (0,1) (0,2) (0,3) (1,0) (1,1) (1,2) (1,3) (2,0) (3,0) (3,1) (3,2) (3,3) (2,1) (2,2) (2,3) ... ... We have put the diagonal, here, in bold, since the subset of N2 consisting of the pairs lying on the diagonal, i.e., {{0,0), (1,1), (2, 2),...}, is the identity relation on N. (Since the identity relation is popular, let's define Id, {{x,x) : x € X} for any set A.) The subset of all pairs lying above the diagonal, i.e., L = {(0,1), (0,2),..., (1,2), (1,3),..., (2, 3), (2,4), .},
is the less than relation, i.e., Lnm iff n < m. The subset of pairs below the diagonal, i.e., G = {(1,0), (2,0), (2, 1), (3,0), (3,1), (3,2),.}, is the greater than relation, i.e., Gnm iffn > m. The union of L with I, which we might call K = LUI, is the less than or equal to relation: Knm iff n < m. Similarly, H = G U I is the greater than or equal to relation. These relations L, G, K, and H are special kinds of relations called orders. L and G have the property that no number bears L or G to itself (i.e., for all n, neither Lnn nor Gnn). Relations with this property are called irreflexive, and, if they also happen to be orders, they are called strict orders. Although orders and identity are important and natural re- lations, it should be emphasized that according to our defini- tion any subset of A² is a relation on A, regardless of how un- natural or contrived it seems. In particular, 0 is a relation on any set (the empty relation, which no pair of elements bears), and A2 itself is a relation on A as well (one which every pair bears), called the universal relation. But also something like E = {(n, m) : n > 5 or m x n > 34} counts as a relation.
Transcribed Image Text:is the less than relation, i.e., Lnm iff n < m. The subset of pairs below the diagonal, i.e., G = {(1,0), (2,0), (2, 1), (3,0), (3,1), (3,2),.}, is the greater than relation, i.e., Gnm iffn > m. The union of L with I, which we might call K = LUI, is the less than or equal to relation: Knm iff n < m. Similarly, H = G U I is the greater than or equal to relation. These relations L, G, K, and H are special kinds of relations called orders. L and G have the property that no number bears L or G to itself (i.e., for all n, neither Lnn nor Gnn). Relations with this property are called irreflexive, and, if they also happen to be orders, they are called strict orders. Although orders and identity are important and natural re- lations, it should be emphasized that according to our defini- tion any subset of A² is a relation on A, regardless of how un- natural or contrived it seems. In particular, 0 is a relation on any set (the empty relation, which no pair of elements bears), and A2 itself is a relation on A as well (one which every pair bears), called the universal relation. But also something like E = {(n, m) : n > 5 or m x n > 34} counts as a relation.
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