Question

3. Consider the relation S, English words having the same number of vowels (a,e,i,o,u). a. Prove it is an Equivalence Relation b. List 3 words that are related to the word 'water' c. Find the equivalence class of (water) and give two elements of [water]

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Answer 1

on w, and We are same number w as w, and 3. @ The relation s is defined the set of all English words as follows: two words related to each other if w, and Wz have of vowels (ae,i,o,u). For all words we have who trivially w and w have same number of vowels. Let w, and we be two words such that W.Swg. Then W2 have same number of vowels. So We and w, also have same number of vowels. Therefore we Sw, holds. So s is symmetric. W., We and Wz be three words such that w, Sw, and Then Wz have same number of rowels. Also we and we have same number of vowels. Therefore w, and was have same number of vowels. and hence w, S W₂ holds. This shows that S is transitive. Hence S is an Let W SW₂ w, and equivalence relation.

In the word 'water' the vowels are a and e. That is 'water has two vowels. 'table has also two vowels. So 'table is related to the word 'water', 'chair' and 'tooth, having two vowels are related to the word 'water'. с The equivalence class of water is. [water] contains the word which are related to the word 'water'. That is [water] contains the words which have 2 vowels. table and chair' are two elements of [water]