Relations And Functions

1. Reflexivity:
A relation on is reflexive if for all , i.e., .
Consider . Then: Hence, . Therefore, is not reflexive.
2. Symmetry:
A relation is symmetric if .
Take . Then: But: Hence, is not symmetric.
3. Transitivity:
A relation is transitive if and .
Let . Then: But: This example does not disprove transitivity. Let's try another one.
Let . Then: But: Hence, is not transitive.
Conclusion:
The relation is neither reflexive, nor symmetric, nor transitive.

Given: defined by , where is the greatest integer less than or equal to .
To prove: is neither one-one nor onto.

1. Not One-One (Injective):
A function is one-one if different inputs give different outputs.
Take two real numbers: and
But and
Therefore, is not one-one.
2. Not Onto (Surjective):
A function is onto if every real number has a pre-image in the domain.
The function always gives an integer as output.
For example, there is no real number such that or , since the range of the function is only integers.
Therefore, all real numbers are not covered in the co-domain .
Hence, is not onto.
Conclusion: The greatest integer function is neither one-one nor onto.