Hi. I am reading Halmos's Naive Set Theory book. I have the following doubts.(adsbygoogle = window.adsbygoogle || []).push({});

1. In chapter 7. on relations, he says "The least exciting relation is the empty one. (To prove that ∅ is a set of ordered pairs, look for an element of ∅ that is not an ordered pair)

". Whenever someone talks about ∅ used in the context of a relation, it becomes very confusing. Formally, is there a way to logically explain as to why ∅ is a relation?

2. At the end of chapter 8., there are two excercises namely "(i) Y^{∅}has exactly one element, namely ∅, whether Y is empty or not, and (ii) if X is not empty, then ∅^{X}is empty". What do these statements mean in the first place? How do you prove them?

I have been able to understand almost everything in Halmos's book except few things which are really confusing. I kindly request you to help me understand them. I really like Halmos's style but I get struck at places and I feel very helpless during those times.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Empty functions and relations

Loading...

Similar Threads - Empty functions relations | Date |
---|---|

B Empty domains and the vacuous truth | Dec 26, 2017 |

Infimum and supremum of empty set | Jul 3, 2015 |

Salmon's 'proof' for the existence of the empty set | Nov 10, 2014 |

The notion of injectivity is undefined on the empty set function? | Mar 10, 2012 |

Empty Function | Jul 28, 2011 |

**Physics Forums - The Fusion of Science and Community**