
An “Anti-symmetric” relation need not be a reflexive relation: give an example.
Answer
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Hint: First we learn the definition of Anti-symmetric relation and reflexive relation. Then we learn about the conditions for a relation to be Antisymmetric relation and reflexive relation. Then we solve the above question by giving an example.
Complete step-by-step answer:
We know that a relation R in mathematics is a connection between the elements of two or more sets.
Suppose we have a set A and if every element of set A maps for itself, then set A is a reflexive relation. It is written as and
For example- we have a set . For reflexive relation following condition must be satisfied- .
Now let’s understand the anti-symmetric relation by an example.
A relation R is antisymmetric if with then must not hold.
For example- we have a relation on the set . This relation is Antisymmetric relation because
Also, the relation is Anti-symmetric relation but it is not reflexive as .
Hence it is proved that an “Anti-symmetric” relation need not be a reflexive relation.
Note: To solve this question students must have a clear idea about Anti-symmetric relation and reflexive relation. Reflexive relations can be symmetric relations. If we have anything other than the same ordered pair in a relation, it will not meet the Antisymmetric requirement.
Complete step-by-step answer:
We know that a relation R in mathematics is a connection between the elements of two or more sets.
Suppose we have a set A and if every element of set A maps for itself, then set A is a reflexive relation. It is written as
For example- we have a set
Now let’s understand the anti-symmetric relation by an example.
A relation R is antisymmetric if
For example- we have a relation
Also, the relation
Hence it is proved that an “Anti-symmetric” relation need not be a reflexive relation.
Note: To solve this question students must have a clear idea about Anti-symmetric relation and reflexive relation. Reflexive relations can be symmetric relations. If we have anything other than the same ordered pair in a relation, it will not meet the Antisymmetric requirement.
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