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Mathematics Question on Functions

Let W denote the words in the English dictionary. Define the relation RR by : R =\\{ (x,y), \in\, W \times \,W : the words xx and yy have at least one letter in commona\\} Then R is

A

reflexive, not symmetric and transitive

B

not reflexive, symmetric and transitive

C

reflexive, symmetric and not transitive

D

reflexive, symmetric and transitive

Answer

reflexive, symmetric and not transitive

Explanation

Solution

Let w∈Ww \in W then (w,w)∈ R ∴ R(w, w) \in \: R \: \therefore \: R is reflexive. Also if w1,w2∈ Ww_1, w_2 \in\,W and (w1,w2)∈ R,  (w_1,w_2) \in\, R,\, then  (w2,w1)∈ R. ∴ R\, (w_2, w_1) \in\, R.\, \therefore\, R is symmetric. Again Let w1=INK,w2w_1 = I N K, w_2 = L I N K, w3w_3 = L E T Then (w1,w2)∈ R(w_1, w_2) \in\, R [∵\because I, N are the common elements of w1,w2](w2,w3)∈ Rw_1, w_2](w_2,w_3)\in\,R [∵\because L is the common element of w2,w3w_2, w_3] But (w1,w3)∉ R(w_1, w_3) \notin \,R [∵\because there is no common element of w1,,w3w_1,, w_3] ∴\therefore R is not transitive.