A relation R on the set A is irreflexive if for every a ∈A,(a, a) ∉R . That is, R is irreflexive

step 1 We're given the definition of an irreflexive relation. And we're asked to determine which relati

A relation R on the set A is irreflexive if for every a...

A relation $R$ on the set $A$ is irreflexive if for every $a \in A,(a, a) \notin R .$ That is, $R$ is irreflexive if no element in $A$ is related to itself.
Which relations in Exercise 6 are irreflexive?

Key Concepts

Binary Relations

A binary relation is a collection of ordered pairs taken from a set, where each pair represents a relationship between two elements of that set. It forms the basis for discussing properties such as reflexivity, symmetry, and transitivity in set theory.

Irreflexivity

Irreflexivity is a property of a relation where no element of the set is related to itself. This means that for every element in the set, the pair in which the element appears twice, (a, a), is not included in the relation.

Set Theory

Set theory underpins the study of relations by providing a framework for dealing with collections of objects. Understanding sets and their elements is essential for grasping how relations, including irreflexive ones, are constructed and analyzed.

Properties of Relations

Properties of relations, such as reflexivity, symmetry, and transitivity, help to classify and understand the behavior of different relations. Identifying whether a relation is irreflexive is a key step in understanding its overall structure and implications.

Transcript

00:01 We're given the definition of an irreflexive relation.

00:08 And we're asked to determine which relations in exercise six are irreflexive.

00:21 So we have that exercise six in part a, the relation r is the set of all real numbers pairs, x, y, such that x plus y is equal to zero.

00:58 That this relation is not irreflexive.

01:06 To see y, we have that 0 is a real number, and we have that 0 plus 0 equals 0.

01:27 In part b, the relation is set of all pairs x, y, such that x is equal to plus or minus y.

01:58 Now, we have that this relation is not irreflexive to cy.

02:08 Notice that again, 0 is a real number, and we have that 0 is equal to 0, which is the same as negative 0.

02:32 In part c, we're given the relation, which is the set of all pairs, x, y, such that x minus y is a rational number.

02:50 We have that this is not an irreflexive relation, and to see y, again, 0 is a real number, and we have that 0 minus 0 is equal to 0, which is an element of the rational numbers, since 0 is equal to 0 over 1.

03:34 In part d, we're given the relation r, which is the set of all pairs x, y, such that x is x is equal to 2y.

03:50 Once again, this relation is not irreflexive.

03:57 To see why, observe that 0 is a real number, and that 0 is equal to 2 times 0, which is 0.

04:13 In part e, we're given the relation r equals the set of pairs xy such that x times y is greater than or equal to 0...

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