Question

Let W denotes the words in the English dictionary. Define the relation \(\mathrm{R}\) by \(\mathrm{R}=\\{(\mathrm{x}, \mathrm{y}) \in \mathrm{W} \times \mathrm{W}\) the ward \(\mathrm{x}\) and \(\mathrm{y}\) 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

Short answer

The relation R is reflexive, symmetric, and not transitive. Therefore, the correct answer is (b) Reflexive, symmetric and not transitive.

Step-by-step solution

Verify if R is reflexive

To check if R is reflexive, we need to see if (x, x) is in R for all x in W. Since any word shares at least one letter with itself, (x, x) must be in R for all x in W. Therefore, R is reflexive.

Verify if R is symmetric

To check if R is symmetric, we need to see if (x, y) being in R implies (y, x) is also in R. If x and y share at least one letter, then it must also be true that y and x share at least one letter. Therefore, (y, x) is also in R. So, R is symmetric.

Verify if R is transitive

To check if R is transitive, we need to see if (x, y) and (y, z) being in R implies (x, z) is also in R. Suppose x and y share the letter A, and y and z share the letter B. If x and z do not share any common letters, then R is not transitive. For example, consider the words "apple" (x), "pear" (y), and "bear" (z). "Apple" and "pear" share the letter "p", and "pear" and "bear" share the letter "e". However, "apple" and "bear" do not have any common letters. Therefore, R is not transitive. So, (a) Not reflexive, symmetric and transitive is incorrect. (b) Reflexive, symmetric and not transitive is correct. (c) Reflexive, symmetric and transitive is incorrect. (d) Reflexive, not symmetric and transitive is incorrect. The answer to the question is: (b) Reflexive, symmetric and not transitive.

Key concepts

Reflexive Property

Understanding the reflexive property in mathematical relations helps you dive deeper into the world of algebra and logic. The reflexive property states that every element relates to itself. Think of it as a person looking into a mirror. Just like their reflection is always there, in terms of a mathematical set or relation, every element 'reflects' back on itself.

For example, let's consider the set of all English words, as given in the exercise problem. If we have a word 'book', the reflexive property means that 'book' must be related to 'book'. Practically speaking, since every word has at least one letter in common with itself, the reflexive property holds true for this relation. That means no matter which word you pick from the English dictionary, it will always satisfy the condition of having at least one letter in common with itself.

This is essential to understand because reflexivity is a foundation for many theories and concepts in mathematics, including equivalence relations and order relations.

Symmetric Property

The symmetric property is another intriguing characteristic of mathematical relations. It's a bit like a two-way street in a neighborhood; if you can go from one house to another, you can just as easily return to where you started. In formal terms, a relation is symmetric if for every pair of elements (x, y) that is related, the pair (y, x) is also related.

Let's apply this to our word relation problem. Imagine two words: 'cat' and 'hat'. They share the letter 'a', so according to our relation, 'cat' relates to 'hat'. The symmetric property assures us that the reverse is also true—'hat' relates back to 'cat'. This symmetry is like a two-way relationship where sharing is mutual. For all words in the dictionary, this property is evident, as letters don't change their identity when the order of words is reversed.

When studying this property, always imagine flipping the elements in question. If the relation still makes sense after the flip, then symmetry holds strong in that relation.

Transitive Property

Moving on to the transitive property, which is a bit like a relay race in mathematics. If the first person passes the baton to the second, and the second person passes it to the third, then there's an indirect connection between the first and third persons. In mathematical terms, a relation is transitive if whenever one element relates to a second, and the second relates to a third, then the first must also relate to the third.

In the context of our word example with the words 'apple', 'pear', and 'bear', we see that 'apple' relates to 'pear', and 'pear' relates to 'bear'. However, because 'apple' and 'bear' do not share any common letter, the chain is broken, and thus the relation is not transitive. This is where our word relation falters. Transitivity would require that any three words with a common link in pairs would need to maintain that linkage across all three, which isn't the case here.

The transitive property is a crucial element of equivalence relations and order relations, and it's important for students to examine elements carefully to determine if this property is present in a given relation or not.