Question

Let W denote the words in the English dictionary. Define the relation R by: R= {(x, y) ϵ W X W the words x and y have at least one letter in common}. Then R is
A.Reflexive, symmetric and not transitive
B.Reflexive, symmetric and transitive
C.Reflexive, not symmetric and transitive
D.Not reflexive, symmetric and transitive

Answer 1

Hint: In this question set R is given where W denotes the words in the English dictionary and has words x and y have at least one letter in common so we will check each of the conditions one by one for x and y.

Complete step-by-step answer:
Given the\[R = \left\{ {\left( {x,y} \right) \in W \times W} \right\}\] , here W denotes the words in the English dictionary
Now since x is a word which belongs in English dictionary so we can clearly say \[\left( {x,x} \right) \in R\] for all \[x \in W\] hence we can say R is a reflexive set.
Now let \[\left( {x,y} \right) \in R\] and\[\left( {y,x} \right) \in R\] , as already said words x and y have at least one letter in common we can say R is symmetric.
Now let\[x = MAT\] , \[y = RAT\] , \[z = RING\]
Hence we can say\[\left( {x,y} \right) \in R\] , \[\left( {y,z} \right) \in R\] but\[x\left( {y,z} \right) \notin R\] , hence we can say R is not transitive.
Therefore R is reflexive, symmetric and not transitive
So, the correct answer is “Option A”.

Note: For a given set X, the set X can be reflexive if each element of the set X is related to itself. It is defined as\[\left( {a,a} \right) \in R\nabla a \in X\] , the set has reflexive property.
If relation X on a set A is symmetric if it holds the \[\left( {a,b} \right) \in X\] and \[\left( {b,a} \right) \in X\] for all \[a,b \in A\] that is \[aRb = bRa\] for all \[a,b \in A\]
The relation X on a set A is said to be transitive if \[\left( {a,b} \right) \in X,\left( {b,c} \right) \in X\] then\[\left( {a,c} \right) \in X\] , such that for all a, b, c ϵA.