Relations

Types of relations on a set: - Suppose R is a relation on a set A. That is R\( \subseteq \)A\( \times \)A. We shall now discuss certain types of relations on a set A

i) Reflexive relation: R is a relation in A and for every a\( \in \)A,(a, a)\( \in \)R, then R is said to be reflexive relation.
Examples:
1.     Every real number is equal to itself. Therefore “ is equal to” is a reflexive relation in the set of real numbers.
2.     If Q is the set of all rational numbers and R is a relation in Q defined by (x, y)\( \in \)R if and only if x < y, then R is not reflexive for x < x for any x\( \in \)Q.
3.     The relation R = {(a, a),(b, b),(c, d),(c, a),(c, c),(d, d)} A = {a, b, c, d} is reflexive for (a, a), (b, b)(c, c),(d, d) \( \in \) R.
4.     l₁, l₂ are two straight lines in a plane. R is a relation in the set of lines in the plane defined by l₁ is parallel to l₂ if (l₁, l₂)\( \in \)R. Then R is reflexive as every line is parallel to itself.
5.     In the set of all triangles in a plane the relation defined by id congruent to is reflexive relation.

ii) Symmetric relation: R is a relation in A and (a, b)\( \in \)R implies (b, a)\( \in \)R then R is said to be symmetric relation.
Examples:
1. In the set of all real numbers “is equal to” relation is symmetric.
2. In set of all natural numbers the relation R defined by x is a factor of y if (x, y)\( \in \) R, Then R is not symmetric for (2, 6)\(\in\)R (since2|6). But 6 is not a factor of 2. \( \therefore \) (6, 2)\( \notin \)R
3. In the set of all triangles in a plane the relation R defined by ‘triangle\(\Delta _1 \) is similar to the triangle\( \Delta _2 \) is symmetric for if \( \Delta _1 \) is similar to \( \Delta _2 \) then \( \Delta _2 \) is similar to \( \Delta _1 \)

iii) Anti symmetric relation: R is a relation in A. If (a, b) \( \in \) R and (b,a)\( \in \)R implies a=b, then R is said to be an anti symmetric relation.
1.     In the set of all natural numbers the relation R defined by x divides y if and only if (x, y) \( \in \) R is anti symmetric. For is x|y and y|x then x= y
2.     A is a set of sets, R is a relation in A defined by “(x, y)  R if and only xy” then R is anti symmetric.
    (x, y), (y, x) \( \in \) R implies that x\( \subseteq \)y and y \( \subseteq \) x, so that x  = y.
\(\therefore\) R is anti symmetric.
3. In the set of all real numbers the relation \( \geqslant \) is an anti symmetric relation. For x\( \geqslant \) y and y \( \geqslant \) x imply x = y for x, y \(\in\) R.
4. The relation R = {(1, 1)(2, 2)(2, 3)} in the set A = {1, 2, 3} is an anti symmetric relation as the condition for anti symmetric relation for the elements of R is trivially true. But it is not reflexive relation because 1, 2, 3 \(\in\) A, but (3, 3)\(\notin\)R though ( 1, 1), (2, 2)\(\in\)R.

iv)Transitive relation: R is a relation in A, If (a, b)\(\in\)R and (b, c)\(\in\)R implies (a, c)  R then R is called a transitive relation
Examples:
1. In the set of real numbers the relation “is equal to” is a transitive relation, For a = b, b = c implies a = c.
2. A is the set of all lines in a plane. R id the relation “is perpendicular to” in A then R is not a transitive relation. For l\( \bot \)m, m\( \bot \)n do not imply l || n in fact l||n. Thus R is not transitive.

v) Equivalence relation: A relation R in a set A is said to be an equivalence relation if it is reflexive, symmetric and transitive.
Example:
1.  T is the set of all triangles in a plane. For x, y \( \in \) T, the relation R is defined by x is congruent to y then R is an equivalence relation for x, y \( \in \) T,
    i) x \( \cong \)x for all x \( \in \)T        ii) x\( \cong \)y, y\( \cong \) z imply x z    iii) x\( \cong \)y imply y\( \cong \) x
2.  In the set of all real numbers the relation “is equal to” is an equivalence relation for a \( \in\) R, a = a, a = b implies b = a and a = b, b = c implies a = c
3.  In the set of positive integers the relation R ‘x divides y” for (x, y) \(\in\) R, is not an equivalence relation for it is not symmetric.
Note: To show that is given relation is not an equivalence relation it is enough to see that one of the properties reflexive, symmetric, transitive is not true.