Let \(S\) be the set of all real numbers and \(R\) be a relation on \(S\) defined by \(xRy \Rightarrow |x| \le y\). Then \(R\) is
transitive only
Reflexivity fails since \(xRx\) requires \(|x| \le x\), which is false for negative \(x\). For transitivity, if \(xRy\) and \(yRz\), then \(|x| \le y\) and \(|y| \le z\); since \(y \ge |x| \ge 0\), we have \(|y| = y \le z\), so \(|x| \le y \le z\), giving \(xRz\). Hence \(R\) is transitive only.
Similar Questions
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:
Let X be the set of all persons living in Delhi. The person's a and b in X are said to be related if the difference in their ages is at most 5 years. The relation is?
Let X be the set of all persons living in a city. Persons x, y in X are said to be related as x < y if y at least 5 years older than x. which one of the following is correct?
Let Z be the set of integers and aRb, where a, b ∈ Z if and only if (a - b) is divisible by 5.
Consider the following statements:
1. The relation R partitions Z into five equivalent classes
2. Any two equivalent classes are either equal or disjoint
Which of the above statements is/are correct?
Let R be a relation defined as xRy if and only if 2x + 3y = 20, where x, y ∈ N. How many elements of the form (x, y) are there in R?
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Let S be the set of all persons living in Delhi. We say that x, y in S are related if they were born in Delhi on the same day. Which one of the following is correct?
Consider the following statements in respect of a relation \(R\) from a set \(A\) to a set \(B\):
I. The set \(B\) is called codomain of the relation \(R\).
II. The range of the relation is always equal to codomain of the relation \(R\).
III. The domain of the relation \(R\) must be equal to the set \(A\).
Which of the statements given above is/are correct?
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:
- Let A be {I, m, n}. Let the relation R be {}. Which of the following statements about R is true?
- 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?
