Let A be {I, m, n}. Let the relation R be {}. Which of the following statements about R is true?
To determine the truthfulness of the statements about the relation R on the set A, we need to analyze its properties: reflexivity, symmetry, and transitivity.
Understanding Relations on a Set
A relation R from a set A to a set B is a subset of the Cartesian product $A \times B$. In this question, the relation R is defined on the set A = {l, m, n}, which means R is a subset of $A \times A$. The given relation R is the empty set, denoted as $R = \emptyset = \{\}$.
Analyzing Reflexivity of Relation R
A relation R on a set A is said to be reflexive if for every element $x \in A$, the ordered pair $(x, x)$ belongs to R.
- The given set is $A = \{l, m, n\}$.
- For R to be reflexive, it must contain the pairs $(l, l)$, $(m, m)$, and $(n, n)$.
- However, the relation R is given as the empty set, $R = \{\}$.
- Since none of the pairs $(l, l)$, $(m, m)$, or $(n, n)$ are present in R, the relation R is not reflexive.
Analyzing Symmetry of Relation R
A relation R on a set A is said to be symmetric if for all elements $x, y \in A$, whenever $(x, y)$ belongs to R, then $(y, x)$ must also belong to R.
- The given relation is $R = \{\}$.
- To check for symmetry, we look for pairs $(x, y)$ in R.
- Since R is the empty set, there are no pairs $(x, y)$ present in R.
- The condition for symmetry is an "if-then" statement. When the "if" part (the premise) is false (i.e., there are no pairs $(x, y)$ in R), the entire "if-then" statement is considered true, regardless of the "then" part. This concept is known as vacuously true.
- Therefore, because there are no elements $(x, y) \in R$ for which the symmetric condition could fail, the relation R is symmetric.
Analyzing Transitivity of Relation R
A relation R on a set A is said to be transitive if for all elements $x, y, z \in A$, whenever $(x, y)$ belongs to R and $(y, z)$ belongs to R, then $(x, z)$ must also belong to R.
- The given relation is $R = \{\}$.
- To check for transitivity, we look for pairs $(x, y) \in R$ and $(y, z) \in R$.
- Since R is the empty set, there are no such pairs $(x, y)$ and $(y, z)$ present in R.
- Similar to symmetry, the condition for transitivity is an "if-then" statement. The "if" part (the premise: $(x, y) \in R$ AND $(y, z) \in R$) is false because R is empty.
- Thus, the entire "if-then" statement is vacuously true.
- Therefore, the relation R is transitive.
Summarizing Properties of Relation R
Let's summarize the findings for the relation R = {} on the set A = {l, m, n}:
| Property | Condition | Result for R = {} |
|---|---|---|
| Reflexive | For all $x \in A$, $(x, x) \in R$ | Not satisfied (e.g., $(l, l) \notin R$) |
| Symmetric | For all $x, y \in A$, if $(x, y) \in R$, then $(y, x) \in R$ | Vacuously true (no $(x, y) \in R$) |
| Transitive | For all $x, y, z \in A$, if $(x, y) \in R$ and $(y, z) \in R$, then $(x, z) \in R$ | Vacuously true (no $(x, y)$ and $(y, z)$ pairs in R) |
Based on this analysis, R is not reflexive, is symmetric, and is transitive.
Important Questions from Types of Relations
A Relation in R is defined as R = {(a, b) : a ≤ b2} is ________.
Let S = {1, 2, 3, ...}, A relation R on S × S is defined by xRy if log ax > log ay when a \(\rm = \frac 1 2.\) Then the relation is:
- Which of the following relations is symmetric but neither reflexive nor transitive for a set A= {a, b, c}?
Consider the following relations on the set {1, 2, 3, 4}:
R1 = {(1, 1),(1, 2), (1, 4),(2, 1), (2, 2), (3, 3),(4, 1), (4, 4)}
R2 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
R3 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3 ), (2, 4), (3, 3), (3, 4), (4, 4)}
R4 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}
Which of these relations are reflexive and transitive but NOT symmetric?
The relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} on a set A = {1, 2, 3} is
