Let W denote the words in the English...

MathematicsTypes of RelationsJEE Main 2006Easy

View in:English Hindi

Let W denote the words in the English dictionary. Define the relation R by $R = {(x, y) \in W \times W ∣ the words x and y have at least one letter in common.}$ Then R is

(A)

not reflexive, symmetric and transitive

(B)

reflexive, symmetric and not transitive

(C)

reflexive, symmetric and transitive

(D)

reflexive, not symmetric and transitive

Answer:B

Visualized Solution (English)

Understanding the Relation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R

Let $W$ be the set of all words in the English dictionary.
Relation $R$ is defined as: $R = {(x, y) \in W \times W ∣ x \cap y \neq = \emptyset}$ , where $x$ and $y$ represent the set of letters in the words.

Checking for Reflexivity

For reflexivity, we check if $(x, x) \in R$ for all $x \in W$ .
Since every word $x$ has at least one letter, $x \cap x = x \neq = \emptyset$ .
Thus, $(x, x) \in R$ , so $R$ is reflexive.

Checking for Symmetry

For symmetry, if $(x, y) \in R$ , then $(y, x)$ must be in $R$ .
$(x, y) \in R ⟹ x \cap y \neq = \emptyset$ .
Since $x \cap y = y \cap x$ , then $y \cap x \neq = \emptyset ⟹ (y, x) \in R$ .
Thus, $R$ is symmetric.

Checking for Transitivity

For transitivity, if $(x, y) \in R$ and $(y, z) \in R$ , then $(x, z)$ must be in $R$ .
Counter-example: Let $x = cat$ , $y = apple$ , $z = pen$ .
$(x, y) \in R$ because they share 'a'.
$(y, z) \in R$ because they share 'p' and 'e'.
But $x \cap z = \emptyset$ , so $(x, z) \in / R$ .
Thus, $R$ is not transitive.

Final Conclusion

The relation $R$ is reflexive and symmetric.
The relation $R$ is not transitive.
Final Answer: Reflexive, symmetric and not transitive.

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let W denote the words in the English dictionary. Define the relation R by R={(x,y)∈W×W∣the words x and y have at least one letter in common.} Then R is

Visualized Solution (English)

Understanding the Relation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R

Checking for Reflexivity

Checking for Symmetry

Checking for Transitivity

Final Conclusion

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