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A relation R is defined on ordered pairs of integers as follows: (x,y) R(u,v) if x < u and y > v. Then R is: Then R is:
- a)Neither a Partial Order nor an Equivalence Relation
- b)A Partial Order but not a Total Order
- c)A Total Order
- d)An Equivalence Relation
Correct answer is option 'A'. Can you explain this answer?
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A relation R is defined on ordered pairs of integers as follows: (x,y)...
An equivalence relation on a set x is a subset of x*x, i.e., a collection R of ordered pairs of elements of x, satisfying certain properties. Write “x R y” to mean (x,y) is an element of R, and we say “x is related to y,” then the properties are:
1. Reflexive: a R a for all a Є R,
2. Symmetric: a R b implies that b R a for all a,b Є R
3. Transitive: a R b and b R c imply a R c for all a,b,c Є R.
An partial order relation on a set x is a subset of x*x, i.e., a collection R of ordered pairs of elements of x, satisfying certain properties. Write “x R y” to mean (x,y) is an element of R, and we say “x is related to y,” then the properties are:
1. Reflexive: a R a for all a Є R,
2. Anti-Symmetric: a R b and b R a implies that for all a,b Є R
3. Transitive: a R b and b R c imply a R c for all a,b,c Є R.
An total order relation a set x is a subset of x*x, i.e., a collection R of ordered pairs of elements of x, satisfying certain properties. Write “x R y” to mean (x,y) is an element of R, and we say “x is related to y,” then the properties are:
1. Reflexive: a R a for all a Є R,
2. Symmetric: a R b implies that b R a for all a,b Є R
3. Transitive: a R b and b R c imply a R c for all a,b,c Є R.
4. Comparability : either a R b or b R a for all a,b Є R.
As given in question, a relation R is defined on ordered pairs of integers as follows: (x,y) R(u,v) if x < u and y > v , reflexive property is not satisfied here, because there is > or < relationship between (x ,y) pair set and (u,v) pair set . Other way , if there would have been x <= u and y>= v (or x=u and y=v) kind of relation among elements of sets then reflexive property could have been satisfied. Since reflexive property in not satisfied here , so given realtion can not be equivalence, partial orderor total order relation.
So, option (A) is correct.
Most Upvoted Answer
A relation R is defined on ordered pairs of integers as follows: (x,y)...
Explanation:
Relation R: R is defined on ordered pairs of integers as follows: (x,y) R(u,v) if x ≤ u and y ≤ v.
Partial Order: A relation R is a partial order if it is reflexive, anti-symmetric, and transitive.
Reflexive: For all a ∈ A, (a,a) ∈ R. (true for this relation)
Anti-symmetric: If (a,b) ∈ R and (b,a) ∈ R, then a = b. (not true for this relation)
Transitive: If (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R. (true for this relation)
Total Order: If a relation R is a partial order and for every two distinct elements a and b in the set A, either (a,b) ∈ R or (b,a) ∈ R, then R is a total order. (not true for this relation)
Equivalence Relation: A relation R is an equivalence relation if it is reflexive, symmetric, and transitive.
Reflexive: For all a ∈ A, (a,a) ∈ R. (true for this relation)
Symmetric: If (a,b) ∈ R, then (b,a) ∈ R. (not true for this relation)
Transitive: If (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R. (true for this relation)
Since the relation R is not anti-symmetric, it cannot be a partial order. Since the relation R is not symmetric, it cannot be an equivalence relation. Therefore, the correct answer is option 'A': Neither a Partial Order nor an Equivalence Relation.
Relation R: R is defined on ordered pairs of integers as follows: (x,y) R(u,v) if x ≤ u and y ≤ v.
Partial Order: A relation R is a partial order if it is reflexive, anti-symmetric, and transitive.
Reflexive: For all a ∈ A, (a,a) ∈ R. (true for this relation)
Anti-symmetric: If (a,b) ∈ R and (b,a) ∈ R, then a = b. (not true for this relation)
Transitive: If (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R. (true for this relation)
Total Order: If a relation R is a partial order and for every two distinct elements a and b in the set A, either (a,b) ∈ R or (b,a) ∈ R, then R is a total order. (not true for this relation)
Equivalence Relation: A relation R is an equivalence relation if it is reflexive, symmetric, and transitive.
Reflexive: For all a ∈ A, (a,a) ∈ R. (true for this relation)
Symmetric: If (a,b) ∈ R, then (b,a) ∈ R. (not true for this relation)
Transitive: If (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R. (true for this relation)
Since the relation R is not anti-symmetric, it cannot be a partial order. Since the relation R is not symmetric, it cannot be an equivalence relation. Therefore, the correct answer is option 'A': Neither a Partial Order nor an Equivalence Relation.
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A relation R is defined on ordered pairs of integers as follows: (x,y) R(u,v) if x < u and y > v. Then R is: Then R is:a)Neither a Partial Order nor an Equivalence Relationb)A Partial Order but not a Total Orderc)A Total Orderd)An Equivalence RelationCorrect answer is option 'A'. Can you explain this answer?
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