## Relations

Question 1 |

R _{1}R_{2} = {(1, 2), (1, 4), (3, 3), (5, 4), (7, 3)} | |

R _{1}R_{2} = {(1, 2), (1, 3), (3, 2), (5, 2), (7, 3)} | |

R _{1}R_{2} = {(1, 2), (3, 2), (3, 4), (5, 4), (7, 2)} | |

R _{1}R_{2} = {(3, 2), (3, 4), (5, 1), (5, 3), (7, 1)} |

Question 1 Explanation:

From the given information,

R

where x+y is divisible by 3

R

where x+y is not divisible by 3

Then the composition of R

(x,z) ∈ R

Thus, R

R

_{1}={(1,2), (1,8), (3,6), (5,4), (7,2), (7,8)}where x+y is divisible by 3

R

_{2}= {(2,2), (4,4), (6,2), (6,4), (8,2)}where x+y is not divisible by 3

Then the composition of R

_{1}with R_{2}denotes R_{1}R_{2}, is the relation from A to C defined by property such as:(x,z) ∈ R

_{1}R_{2}, iff if there is a y ∈ B such that (x,y) ∈ R_{1}and (y,z) ∈ R_{2}.Thus, R

_{1}R_{2}= {(1,2), (3,2), (3,4), (5,4), (7,2)}Question 2 |

Let R be a symmetric and transitive relation on a set A. Then

R is reflexive and hence an equivalence relation | |

R is reflexive and hence a partial order
| |

R is reflexive and hence not an equivalence relation | |

None of the above |

Question 2 Explanation:

If a relation is equivalence then it must be

i) Symmetric

ii) Reflexive

iii) Transitive

If a relation is said to be symmetric and transitive then we can't say the relation is reflexive and equivalence.

i) Symmetric

ii) Reflexive

iii) Transitive

If a relation is said to be symmetric and transitive then we can't say the relation is reflexive and equivalence.

Question 3 |

Amongst the properties {reflexivity, symmetry, anti-symmetry, transitivity} the relation R = {(x,y) ∈ N

^{2}| x ≠ y } satisfies __________symmetry |

Question 3 Explanation:

It is not reflexive as xRx is not possible.

It is symmetric as if xRy then yRx.

It is not antisymmetric as xRy and yRx are possible and we can have x≠y.

It is not transitive as if xRy and yRz then xRz need not be true. This is violated when x=x.

So, symmetry is the answer.

It is symmetric as if xRy then yRx.

It is not antisymmetric as xRy and yRx are possible and we can have x≠y.

It is not transitive as if xRy and yRz then xRz need not be true. This is violated when x=x.

So, symmetry is the answer.

Question 4 |

The less-than relation, <, on reals is

a partial ordering since it is asymmetric and reflexive | |

a partial ordering since it is antisymmetric and reflexive | |

not a partial ordering because it is not asymmetric and not reflexive | |

not a partial ordering because it is not antisymmetric and reflexive
| |

none of the above |

Question 4 Explanation:

Relation < is:

1) not reflexive

2) irreflexive

3) not symmetric

4) Asymmetric

5) Antisymmetric

1) not reflexive

2) irreflexive

3) not symmetric

4) Asymmetric

5) Antisymmetric

There are 4 questions to complete.