relation and operations power point presentation
- 1. Relations and Operations Prepared by: MARIELA A. CAMBA
- 2. Concepts of Relations • Theory of relations is introduced by Augustus De Morgan. We deal with relational operators, namely, ≤, <, >, ≥, ≠, and = are commonly used defining the relations. • In set theory, the concept subset represents the relationship between the two sets. DEFINITION 2.1: A relation R on a set S (more precisely, a binary relation on S, since it will be a relation between pairs of elements of S) is a subset of SxS.
- 3. Cartesian Product • Let A and B be two sets. The ordered pairs of the type (a,b) are obtained in the new set by taking Cartesian product of the given two sets. • A x B = {(a,b) | a ε A and b ε B} • Example: Consider A = { 1, 2, 3} and B = {x, y}. • Obtain the Cartesian product set A x B. • A x B = {( 1, X), ( 1, y), ( 2, X), ( 2, y), ( 3, X), (3, y)}.
- 4. Identity Relation • It is also called Equality relation, wherein every element of the set is related to itself. If A is the set, then the identity relation IA is given as under: • IA = {(a, a) | for all a ε A } • Given a set A = {1,2,3} the relation IA is {(1,1), (2,2),(3,3)}.
- 5. Inverse Relation • Let R be a relation from set A to set B, R: A B. the inverse relation of R is denoted as R -1 and represents the relation B to A. it is defined as under. • • This is equivalent to saying that ( a R b ) if and only if (b R -1 a ). It is also equivalent to saying that (a, b) ε R (b, a) R -1 . The domain of R is identical to the range of R -1 and vice versa. Consider a set A = { 1,2,3} and the relation R = {(1,2), (2,3), (3,3)}. The relation= {(2,1), (3,2), (3,3)}.
- 6. Properties of Binary Relations • In computer science, most often we deal with relations on a given set rather than the relation across two sets. These relations satisfy certain properties. The properties include Reflexivity, Irreflexivity, Symmetry, Anti-symmetry and Transitivity. Reflexivity and Irreflexivity • The relation R on a set A is said to be reflexive if a is related to a (a R a) for all a ε A. Similarly, the relation R on a set A is said to be irreflexive if a is not related to a (a R a) for all a ε A.
- 8. Symmetry, not symmetric and asymmetry • A relation R on a set A is said to be symmetric, whenever • a R b then b R a. The relation R is said to be not symmetric whenever we have some a and b belonging to A and (aRb) but b is not related to a, (bRa) is not true. The relation R is said to be asymmetric, whenever aRb for all a,b ε A, b is not related to a. These are defined as under. • (a, b) ε R (b, a ) ε R for a, b ε A, the relation is symmetric • (a, b) ε R and (b, a ) R for a, b ε A, the relation is not symmetric • (a, b) ε R (b, a ) R for a, b ε A, the relation is asymmetric
- 9. Consider the relation parallel defined over a set of lines. If line A is parallel to line B and line B is parallel to C then the line A is parallel to C. Therefore, the relation parallel is transitive. The same is not true for the relation perpendicular. A relation R is said to be transitive, if (aRb) and (bRc ) implies that (aRc). In other words, the relation is said to be not transitive if (aRb) and (bRc ) but (aRc). EXAMPLES Consider the sets A, B and C and the relation subset defined over a universal set. If A B and B C then A C. Therefore, the relation is transitive.
- 10. Equivalence Relation and Partitions • A given relation is said to be an Equivalence relation when it is reflexive, symmetric and transitive. Example : • Consider the set of triangles in a plane. The relation similarity of triangles is an equivalence relation. Let A,B and C represent the triangles in a plane. From the property of similarity triangles, we can ascertain the following: • A, B and C are similar to them selves. Reflexive property • If triangle A is similar to triangle B then triangle B is similar to triangle A. symmetric property. • If triangle A is similar to triangle B and triangle B is similar to triangle C then triangle A is similar to triangle C. Transitive property.
- 11. Partitions
- 16. Matrix representation of relations
- 17. Types of Binary operations • We take the set of numbers on which the binary operations are performed as X. The operations (addition, subtraction, division, multiplication, etc.) can be generalized as a binary operation is performed on two elements (say a and b) from set X. The result of the operation on a and b is another element from the same set X. • Thus, the binary operation can be defined as an operation x which is performed on a set A. The function is given by x: A x A → A. So the operation x performed on operands a and b is denoted by a x b.
- 18. •Let us show that addition is a binary operation on real numbers (R) and natural numbers (N). So if we add two operands which are natural numbers a and b, the result will also be a natural number. The same holds good for real numbers. Hence, •Let us show that multiplication is a binary operation on real numbers (R) and natural numbers (N). So if we multiply two operands which are natural numbers a and b, the result will also be a natural number. The same holds good for real numbers. Hence, Examples
- 19. •Let us show that subtraction is a binary operation on real numbers (R). So if we subtract two operands which are real numbers a and b, the result will also be a real number. The same does not hold good for natural numbers. It is because if we take two natural numbers, 3 and 4 as a and b, then 3 – 4 = -1, which is not a natural number. Hence, •Similarly, the division cannot be defined on real numbers. This is because / : R x R R is → given by (a, b) aa/b. → Now if we take b as 0 here, a/b is not defined. Examples
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