1. Let A the set of all phone numbers for people living in New York State. Let R
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Question
1. Let A the set of all phone numbers for people living in New York State. Let R be the relation on A where (a, b) E R if and only if a and b have the same area code. Is this relation reflexive? Symmetric? Anti-symmetric? Transitive? Explain your answers.
2. Which of the following relations in 0,1,2,3 are equivalence relations? If the given relation is an equivalence relation, then verify the properties of equivalence relation. If the given relation is not an equivalence relation, then show which property or properties fail.
(a.) {(0,0),(1,1),(2,2),(3,3),(1,2)}
(b.) {(0,0),(1,1),(2,2),(1,2),(2,1)}
(c.) {(0,0),(0,2),(1,1),(2,0),(2,2),(2,3),(3,2),(3,3),(1,2),(2,1)}
(d.) {(0,0),(0,1),(0,3)(1,0),(1,1),(2,2),(2,3),(3,3)}
3. Which of the following collections of subsets of S = {1,2,3,4,5,6} form a partition of S?
If the given collection of sets does form a partition, then list the ordered pairs if the equivalence relation produced by the partition. If the given collection does not form a partition, then explain which part(s) of the definition of partition fails.
(a.) {1,3,5},{2,4}
(b.) {1,2},{3,4},{5,6}
(c.) {1},{2},{5},
(d.) {1,2,3,4},{4,5,6}
Explanation / Answer
1. The relation is reflexive, Symmetric and transitive,i.e., it is an equivalence relation
Reflexivity :- (a,a) E R because the area code of a number will be unique and cannot be different from itself.
Symmetry :- If(a,b) E R, then it means area code of b is same as area code of a which is same as saying area code of a is same as are code of b which means (b,a) E R whenever (a,b) E R
Transitivity :- If (a,b) E R and (b,c) E R, then it means the are code of a is same as that of b and area code of b is same as that of c, which means are code of all a,b,c are same. So, (a,c) E R if(a,b) E R and (b,c) E R
2. a => not an equivalence relation as it does not satisfy symmetry. (1,2) E R but (2,1) does not belong to R which is a violation to symmetry
b => it is an equivalence relation as it satisfy all thre properties of an equivalence relation
Reflexivity:- all of (0,0) , (1,1) and (2,2) belong to R satisfying reflexivity
Symmetry:- (1,2) E R and (2,1) also belong to R. The other three are symmetric so they are symmetric anyways.
transitivity:- (1,2) E R and (2,1) E R. For transitivity (1,1) should belong to R and it does belong to R. So, its transitive
c => Not an equivalence relation as it does not satisfy transitivity. (1,2) E R , (2,3) E R but (1,3) does not belong to R thereby violating transitivity condition.
d => Not an equivalence elation as it does not satisfy symmetry. (0,3) E R but (3,0) does not belong to R
3. Only b part forms a partition as the others parts dont partition the set such that every element occurs only once in either set.
(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(1,3),(1,4),(1,5),(1,6)...... and so on till all elements are there in the set to form equivalence relation
4. a=> It is an Equivalence relation as it satisfies all three properties of eqivalence. The set represents a curve for which z = x + 2y. All points x + 2y lie on this curve.
b => It is an Equivalence relation as it satisfies all three properties of eqivalence.The set represents a curve for which z = 3xy. All points 3xy lie on this curve.
c => Not an equivalence relation as it doesn't satisfy symmetry. For symmetry (a,b) ~ (b,a) E R but for that a + 2b = b + 2a which need not be true.
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