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- 1. Relation
- 2. Representing sets usingcomputer• 1 = true , 0 = false• 10 1010 1010 = {1,3,5,7,9}• Union = OR• Intersection = AND• Complement = inverse all
- 3. Concept of relation between twosets• If we want to describe a relationship between elements of two sets A and B, we can use ordered pairs with their first element taken from A and their second element taken from B.• Since this is a relation between two sets, it is called a binary relation.• Definition: Let A and B be sets. A binary relation from A to B is a subset of A B.• In other words, for a binary relation R we have R A B.• We use the notation aRb to denote that (a, b) R and aRb to denote that (a, b) R.• When (a, b) belongs to R, a is said to be related to b by R.
- 4. example• Example: Let P be a set of people, C be a set of cars, and D be the relation describing which person drives which car(s). P = {Carl, Suzanne, Peter, Carla} C = {Mercedes, BMW, tricycle} D = {(Carl, Mercedes), (Suzanne, Mercedes), (Suzanne, BMW), (Peter, tricycle)}• This means that Carl drives a Mercedes, Suzanne drives a Mercedes and a BMW, Peter drives a tricycle, and Carla does not drive any of these vehicles.
- 5. Properties of relations• Reflexive• A relation R on a set A is called reflexive if (a, a) R for every element a A.• Example:• R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}• Answer: R is reflexive because they both contain all pairs of the form (a, a),namely, (1, 1), (2, 2), (3, 3), and (4, 4).
- 6. • Irreflexive• A relation on a set A is called irreflexive if (a, a) R for every element a A.• Example: These 4 irreflexive relations are : 1. Empty 2. {(1,2)} 3. {(2,1)} 4. {(1,2), (2,1)}
- 7. • Symmetric• A relation R on a set A is called symmetric if (b, a) R whenever (a, b) R for all a, b A.• Example:• R={(1, 1), (1, 2), (2, 1)}• Answer: R is symmetric.Both (2, 1) and (1, 2) are in the relation so R is symmetric.
- 8. • Antisymmetric• A relation R on a set A is called antisymmetric if a = b whenever (a, b) R and (b, a) R.• Example:• R={(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}• Answer:R is antisymmetric.There is no pair of elements a and b with a ≠ b such that both (a, b) and (b, a) belong to the relation.
- 9. • Transitive• A relation R on a set A is called transitive if whenever (a, b) R and (b, c) R, then (a, c) R for a, b, c A.• Example: Are the following relations on {1, 2, 3, 4} transitive?• Answer: R = {(1, 1), (1, 2), (2, 2), (2, 1), (3, 3)}
- 10. Definition or concepts of functionon general sets •In discrete mathematics ,functions are used •in the definition of such discrete structures as sequences and strings. •used to represent how long it takes a computer to solve problems of a given size. Concept of function: •Let A and B be nonempty sets. •A function f from A to B is an assignment of exactly one element of B to each element of A. •We write f (a) = b if b is the unique element of B assigned by the function f to the element a of A. •If f is a function from A to B, we write f : A → B
- 11. •If f is a function from A to B, we say that A is the domain of f and B isthe codomain of f.•If f (a) = b, we say that b is the image of a and a is a preimage of b.•The range, or image, of f is the set of all images of elements of A.,• if f is a function from A to B, we say that f maps A to B.
- 12. Concept of Boolean Function• Definition 1 : A literal is a Boolean variable or its complement. A minterm of Boolean variables x1, x2, . . . , xn is Boolean product y1 y2. . .yn where yi = xi orExample1:Find the minterm F such that F = 1 if x1 = x3 = 0 and F = 0 ifx2 = x4 = x5 = 1.Solution: The minterm isThe sum of minterms that represents the functions iscalled the sum-of-products expansion or the disjunctivenormal form of the Boolean function.
- 13. Example2: Find the sum-of-product expansion for thefunction .(1) Solution:
- 14. Injective functionsInjective means that every member of “A” has its own uniquematching member in “B”.A function f is injective if and only if whenever f(x) = f(y), x =y.It’s also called “one to one”.Ex: {a, b, c, d} to {1,2,3,4,5} with f(a) =2, f(b) =1, f(c)=3, f(d)=4 isone to one. a 1 b 2 c 3 d 4 5
- 15. Surjective functions (an onto function)Surjective means that every “B” has at least one matching “A” (maybe more than one).A function f (from set A to B) is surjective if and only for every y in B, there at least one x in A such that f(x) = y, in other words f is surjective if and only if f(A) = B.Ex: f be the function from {a,b,c,d,e} to {1,2,3,4} defined by f(a)=2, f(b)=1, f(c)=3, f(d)=3, f(e)=4. Is f is onto function?Answer: NO a 1 b 2 c 3 d 4 e
- 16. Bijective functionsBijective means both Injective and Surjective together.Perfect “one-to-one correspondence” between the members of the sets is existed.A function f (from set A to B) is bijective if, for every y in B, there is exactly one x in A such that f(x) = y.Ex: f be the function from {a,b,c,d,e} to {1,2,3,4,5} with f(a)=2, f(b)=1,f (c)=6, f(d)=3, f(e)=5. Is f a bijection?Answer: YES. a 1 b 2 c 3 d 4 e 5
- 17. Definition and example of inversefunction An inverse function, which we call f-1 is another function that take y back to x. f(x)= y. So, f-1(y)= x.
- 18. • Example: Let f: Z Z be such that f(x)=x+1 f(x)= x+1 y=x+1 y-1=x f-1(y)= y-1
- 19. Definiton and example ofcomposition function• Let g be a function from the set A to the set B and let f be a function from the set B to the set C. The composition of the functions f and g, denoted by f ◦ g, is defined by (f ◦ g)(a) = f (g(a)).
- 20. • Example: f (x) = 2x + 3 and g(x) = 3x + 2• Solution: (f ◦ g)(x) = f (g(x)) f (3x + 2) = 2(3x + 2) + 3 = 6x + 7 and (g ◦ f )(x) = g(f (x)) g(2x + 3) = 3(2x + 3) + 2 = 6x + 11.

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