a) Consider the whole of the English words set. Suppose an English word x is related to another English word y if x and y begin with the same letter. i) Show that this is an equivalence relation. ii) Compute C(quadratic) and C(rhombus) iii) How many equivalence classes are there in all, and why? iv) What is the partition of the English words under this relation? b) Consider Z, the set of integers. Suppose we define the relation: x is related to y if x - y > 3, x, y \in Z. Determine whether or not the relation is i) reflexive ii) symmetric iii) transitive
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is related to if and begins with the same letter.
Thus, the relation is reflexive since every English words will be related to itself.
Also, if , that is they both have the same first letter, then Thus the relation is symmetric.
Lastly, if and that is, and have the same letter and and have the same letter. Definitely, and will have the same letter, that is . Thus, the relation is transitive.
Since the relation is reflexive, symmetric and transitive, then the relation is an equivalence relation.
Since there are 26 English alphabets, then will we have 26 equivalence class
The English words will be partition into words with same first alphabet
To check if it is reflexive, let . We want to check if
Thus, it is not reflexive.
To check if it is symmetric. Let and
Thus, the relation is not symmetric.
To check if it is transitive. Let
Add the two together, we have that:
Thus the relation is transitive.