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Discrete Mathematics - Languages and Machines
Need to solve Finite State Automaton i.e,
Your task is to design a binary finite state automaton (FSA) to accept all strings that represent valid messages (for your particular codes and parity property) and reject all others.
This FSA must be DETERMINISTIC, REDUCED and must be in STANDARD FORM.

Q : 1) Construct the call graph for a set of seven telephone
numbers 555-0011, 555-1221, 555-1333, 555-8888,
555-2222, 555-0091, and 555-1200 if there were three
calls from 555-0011 to 555-8888 and two calls from
555-8888 to 555-0011, two calls from 555-2222 to
555-0091, two calls from 555-1221 to each of the
other numbers, and one call from 555-1333 to each of
555-0011, 555-1221, and 555-1200.

Q : 2) Find the sum of the degrees of the vertices of each graph
in Exercises 1–3 and verify that it equals twice the number
of edges in the graph.

Q : 3) Draw these graphs.
a) K7 b) K1,8 c) K4,4
d) C7 e) W7 f) Q4
Q : 5). If f is function from A to B and g is function B to C and both f and g are onto.
Show that go f is also onto. Is go f one-to-one if both f and g are one-to-one.


Q : 6) Let f, g and h: R → R be defined by (R is the set of real numbers)
f(x) = x + 2, g(x) = (1 + x2)
-1, h(x) = 3.
Compute f -1g(x) and hf (g f -1) (hf(x)).
Q : 1) Determine whether each function is one-to-one. The domain of each
function is the set of all real numbers. If the function is not one-to-one,
prove it. Also, determine whether f is onto the set of all real numbers. If f is
not onto, prove it.
a) f(x) = 6x – 9
b) f(x) = 2x3 - 4

Q : 2) Let A = {1, 2, 3}, B = {p, q} and C = {a, b}. Let f: A → B is f = {(1, p), (2, p), (3,
a)} and g: B → C is given by {(p, b), (q, b)}. Find go f and show it pictorially.
Q:1. Determine whether each set is a function from X = {1,2,3,4} to Y = {a,b,c,d}.
If it is a function, find its domain and range, draw its arrow diagram, and
determine if it is one-to-one, onto or both.
a) {(1,a,),(2,a),(3,c),(4,b)}
b) {(1,c),(2,a),(3,b),(4,c),(2,d)}
c) {(1,d),(2,d),(4,a)}

Q:2. List all possible functions from A to B, A = {a, b, c}, B = {0, 1}. Also indicate in
each case whether the function is one-to-one, is onto and one-to-one-onto.
Let R = {(2, 2), (2, 4), (3, 3), (3, 6), (3, 12), (4,2), (6, 3)}
where if a is a factor b}

(i) Draw the digraph for the relation P.
(ii) Find the domain and range of the relation.
Describe two situations that can be modelled with a Graph. Describe the graph model that you will use.
Describe two situations that can be modelled with a Tree. Describe the tree model that you will use.
Describe two situations that can be modelled with a tree . Describe the graph model that you will use
Describe two situations that can be modelled with a Tree. Describe the graph model that you will use.
Describe two situations that can be modelled with a Graph. Describe the graph model that you will use
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