Question

28. Define an "onto" function. Give an example of an onto function from X = {1,2,3,4,5,6,7,8,9,0} onto Y = {a,b,c,d} 29. What is the domain and co-domain of the function which assigns to each pair of non- negative integers the first integer of the pair? What is the range? 30. Let S be the set of all positive integers not exceeding 100. Let f be the function which assigns the cardinality to each subset of S. What is the domain, co-domain and range of the function?

This is discrete mathematics.

I need answer and explanation for #28 #29 #30.

Thank you so much!!

Answer 1

28. Onto function : A function f : A$\rightarrow$B is said to be onto if for each y belongs to B there exist a x belongs to A such that f(x) = y .

X = { 1 , 2 , 3 , 4 , 5 , 6 ,7 ,8 ,9 ,0 } and Y = { a , b , c ,d }

f : X$\rightarrow$Y is defined by ,

f(1)= a

f(2) = b

f(3) = c

f(4) = f(5) = f(6) = f(7)= f(8)=f(9)=f(0) = d

Then each element of Y has a preimage under f . Hence f is onto.

(29) . Let f be the function which takes pair of non-negative integers to the first integer i e., the function is defined by ,

f( a ,b) = a

As f operates on the set of all pair of non-negative integer so domain is = $\mathbb{Z}_{+} \times \mathbb{Z}_{+}$ where $\mathbb{Z}_{+}$ represents set of all non-negative integers .

Also for each non-negative integer a , f ( a, 1 ) = a so for all a has a pre-image .

Hence range is set of all non-negative integer i.e., $\mathbb{Z}_{+}$ .

(30) . S be the set of all positive integer not exceeding 100 . So ,

S = { 1 , 2 , 3 , 4 , 5 , .....99, 100 }

Now f takes a subset of S to its cardinality . So f operate on subsets of S . Hence domain of f is set all of subsets of S i.e., power set of S , .

P(S)

As S contains 100 elements so all its subsets contains elements from 0 to almost 100. So cardinality of subsets of S varies from 0 to 100.  

Hence range of f is { 0 , 1 , 2 , 3 , 4 , 5 , ....,99 , 100} .

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