1. Write precise mathematical definition for the following terms: d) Simple graphe) Equivalence classf) Reflexive relation

Consider two sets in ℝ² with the usual topology. Is the closure of a path-connected set also path-connected? Is the inverse true? (That is to say, if the closure of a set is path-connected, then is the set path-connected?)

For (a)and (b), use an Euler diagram to determine whether the syllogism is valid or invalid. Clearly define and label any sets or elements used with variables.

Consider sets in R^2 with the usual topology. Is the closure of a set connected by paths connected by paths? does the reverse hold? (That is, if the closure of the set is path-connected, is the set path-connected?)   Please argue your answer and give definitions if necessary. Thank you

Suppose A and B are two finite sets such that |B| = 2 |A| and the number of relations on A is 16. Determine: |B|:  Number of relations from A to B: