Function and Relation.pdf
- 1. Function and Relation Prepared by: Angela Clarito
- 2. Relation ● It is a Rule that pairs each element in one set. ● It is a set of ordered pairs, x and y. ● A relation R on a set X is a subset of X x X. If (a,b) ∊ R, we write x R y. Reads as “ x is related to y”.
- 3. Relation ● A = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)) X y 2 1 1 2 3 R
- 6. Reflexive Relation ● A relation R on a set is called reflexive if (a,a) ∊ R for every element a ∊ A. In other words, ∀ a((a,a) ∊ R). ● A relation will be known as reflexive relative if every element of set A is related to itself. The word reflexive means that in a set, the image of every element has its own reflection. Example: Let A = {1,2,3,4} R={(1,1),(2,2),(3,3),(4,4)}
- 7. Irreflexive Relation ● A relation R on a set A is called irreflexive if ∀a ∊ A, (a,a) ∉ R. Example: Let A = {1,2,3,4} R={(1,2),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,2), (3,4),(4,1),(4,2),(4,3)}
- 8. Symmetric Relation ● A relation R on a set A is called symmetric if (b,a) ∊ R holds when (a,b) ∊ R for all a, b ∊ A. ● In other words, relation R on a set A is symmetric if ∀a ∀b((a,b) ∊ R → (b,a) ∊ R. Example: Let A = {1,2,3,4} R={(1,2),(2,1),(1,3),(3,1),(1,4),(4,1)}
- 9. Antisymmetric Relation ● A relation R on a set A is called antisymmetric if ∀a ∀b((a,b) ∊ R ^ (b,a) ∊ R → (a=b)) whenever we have (a,b) in R, we will never have (b,a) in R until or unless (a=b). Example: Let A = {1,2} R1 ={(1,1),(2,1),(2,2)} R2 ={(1,1),(1,2),(2,1),(2,2)}
- 10. Asymmetric Relation ● A relation R on a set A is called asymmetric if ∀a ∀b((a,b) ∊ R ^ (b,a) ∉ R) whenever we have (a,b) in R, we will never have (b,a) in R until or unless (a=b). Example: Let A = {1,2,3,4} R={(1,2),(1,3),(1,4),(2,3),(2,4)}
- 11. Transitive Relation ● A relation R on a set A is called transitive if ∀a ∀b ∀c(((a,b) ∊ R ^ (b,c) ∊ R) → (a,c) ∊ R) Example: Let A = {1,2,3} R={(1,3),(1,2),(3,2)}
- 12. Function ● It is a Rule that pairs each x-coordinate to exactly one element from y-coordinates. ● It is a set of ordered pairs (relation) where x-coordinate should not be repeated. ● Are sometimes called as mappings or transformations. ● A function f from A to B , denoted f: A → B, assign each element of A exactly on element of B.
- 13. Relation ● A = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)) X y 2 1 1 2 3 R
- 14. Terminologies
- 16. Domain Is the set x. Codomain Is the set y . Range The values in set y that is paired. .
- 17. Preimage The value in y that is paired to the value in x. Image The value in x that is paired to the value in y.
- 18. Answer the following for f: X → Y Domain Codomain Range Image(s) Preimage f(y) = ?
- 21. Injective (One to one) Functions Each Value in the range corresponds to exactly one element in then domain. ∀ a ∀ b (( a ≠ b) → f (a) ≠ f (b))) ∀ a ∀ b ((f (a) = f (b)) → (a = b)) ∀ reads as “for all” ≠ reads as “not equal to” CONTRAPOSITIVE
- 22. Injective (One to one) Functions
- 23. Surjective (Onto) Functions Each element in the codomain maps to at least one element in the domain. ∀ y ∃ x (f(x) = y) ∃ reads as “ there exists”
- 25. Bijective (One to one correspondence) Functions Functions that are both one-to-one and onto, or both surjective and injective. ∀ y ∃ x (f(x) = y) ∃ reads as “ there exists”
- 26. Bijective (One to one correspondence) Functions
- 27. Problem: Let f be a function from X = { a, b, c, d } to Y = { 1, 2, 3 } defined by f(a) = 3, f(b) = 2, f(c) = 2 and f(d) = 3. Is f: X → Y either one-to one or onto?
- 28. Examples of Different Types of Correspondences
- 29. One-to-one, not onto Onto, not one-to-one
- 30. One-to-one, onto Neither one-to-one, onto
- 31. Not a Function
- 32. Inverse Function ● Given any function, f, the inverse of the function f-1, is a relation that is formed by interchanging each (x, y) of f to a (y, x) of f-1, . ● The function f would be denoted as f-1, and read as “f inverse”. NOTE: A one-to-one correspondence is called invertible because we can define an inverse of this function. A function is not invertible if it is not a one-to one correspondence, because inverse of such a function does not exist.
- 33. Inverse Function
- 34. Composition of Function ● The composition of two functions g: A → B and f: B → C, denoted by f о g, is defined by (f о g) (a) = f(g(a)) or (g о f) (a) = g(f(a)) This means that First, function g is applied to element A, mapping it into an element of B. Then, function f is applied to this element of B, mapping it into an element of C. Therefore, the composite function maps from A to C.
- 36. Thank You!