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Discrete Mathematics
Let X = {a, b, c} defined by f : X X such that f = {(a, b), (b, a), (c,c)}. Find the values of
f–1, f2 and f4.
b) Let L = {3, 4, 12, 24, 48, 82} and the relation < be defined on L such that x < y if x divides y. Draw the Hasse diagram.
c) Show that the functions, defined by : are inverse of one another.
f–1, f2 and f4.
b) Let L = {3, 4, 12, 24, 48, 82} and the relation < be defined on L such that x < y if x divides y. Draw the Hasse diagram.
c) Show that the functions, defined by : are inverse of one another.
Discrete Mathematics
Let X = {1,2,3,4,5,6,7} and R = {x,y/x–y is divisible by 3} in x. Show that R is an equivalence relation.
b) Let A = {1,2,3,4} and let R = {(1,1), (1,2),(2,1),(2,2),(3,4),(4,3), (3,3), (4,4)} be an equivalence relation on R. Determine A/R.
c) Draw the Hasse diagram of lattices, (L1,<) and (L2,<) where L1 = {1, 2, 3, 4, 6, 12} and
L2 = {2, 3, 6, 12, 24} and a < b if and only if a divides b.
b) Let A = {1,2,3,4} and let R = {(1,1), (1,2),(2,1),(2,2),(3,4),(4,3), (3,3), (4,4)} be an equivalence relation on R. Determine A/R.
c) Draw the Hasse diagram of lattices, (L1,<) and (L2,<) where L1 = {1, 2, 3, 4, 6, 12} and
L2 = {2, 3, 6, 12, 24} and a < b if and only if a divides b.
Discrete Mathematics
Determine whether the following relations are injective and/or subjective function. Find universe of the functions if they exist.
i. A= v,w,x,y,z, B=1,2,3,4,5
R= (v,z),(w,1), (x,3),(y,5)
ii. A = 1,2,3,4,5 B=1,2,3,4,5
R = (1,2),(2,3),(3,4),(4,5),(5,1)
b) If a function is defined as f(x,n) mod n. Determine the
i. Domain of f ii.Range of f iii. G(g(g(g(7)))) if g (n) = f(209, n).
18. a) Let L be a lattice. Then prove that a b=a if and only if a v b=b.
b) Define the dual of a statement in a lattice L. why does the principle apply to L.
i. A= v,w,x,y,z, B=1,2,3,4,5
R= (v,z),(w,1), (x,3),(y,5)
ii. A = 1,2,3,4,5 B=1,2,3,4,5
R = (1,2),(2,3),(3,4),(4,5),(5,1)
b) If a function is defined as f(x,n) mod n. Determine the
i. Domain of f ii.Range of f iii. G(g(g(g(7)))) if g (n) = f(209, n).
18. a) Let L be a lattice. Then prove that a b=a if and only if a v b=b.
b) Define the dual of a statement in a lattice L. why does the principle apply to L.
Discrete Mathematics
Let A,B,C єR2 where A = { (x,y) / y = 2x + 1} , B = { (x,y) / y = 3x} and C = { (x ,y) / x - y = 7} . Determine each of the following:
i. A B ii. B intersection C complement 3.
i. A B ii. B intersection C complement 3.
Discrete Mathematics
13. a) Consider f; Z+ → Z+ define by f(a) =a2. Check if f is one-to-one and / or into using suitable explanation.
b) What is a partial order relation? Let S = { x,y,z} and consider the power set P(S) with relation R given by set inclusion. Is R a partial order.
c) Define a lattice. Explain its properties.
14. Show that if eight people are in a room, at least two of them have birthdays that occur on the same day of the week.
15. a) Give a relation which is both a partially ordered relation and an equivalence relation on a set.
b) Let P be the power set of {a, b, c}. Draw the diagram of the partial order induced on P by the lattice (P,,).
16. a) Let A,B,C єR2 where A = { (x,y) / y = 2x + 1} , B = { (x,y) / y = 3x} and C = { (x ,y) / x - y = 7} . Determine each of the following:
i. A B ii. iii. iv.
b) What is a partial order relation? Let S = { x,y,z} and consider the power set P(S) with relation R given by set inclusion. Is R a partial order.
c) Define a lattice. Explain its properties.
14. Show that if eight people are in a room, at least two of them have birthdays that occur on the same day of the week.
15. a) Give a relation which is both a partially ordered relation and an equivalence relation on a set.
b) Let P be the power set of {a, b, c}. Draw the diagram of the partial order induced on P by the lattice (P,,).
16. a) Let A,B,C єR2 where A = { (x,y) / y = 2x + 1} , B = { (x,y) / y = 3x} and C = { (x ,y) / x - y = 7} . Determine each of the following:
i. A B ii. iii. iv.
Discrete Mathematics
If R is a relation defined on the set Z by a R b if a-b is a non negative even integer. Determine if R define a partial order and total order.
Discrete Mathematics
Let f and g be functions from the positive real numbers to positive real numbers defined by f(x) = [2x]
g(x) = x2. Calculate f o g and g o f.
g(x) = x2. Calculate f o g and g o f.
Discrete Mathematics
Define a bijective function. Explain with reasons whether the following functions are bijective or not. Find also the inverse of each of the functions.
i. f(x) = 4x+2, A=set of real numbers
ii. f(x) = 3+ 1/x, A=set of non zero real numbers
iii. f(x) = (2x+3) mod7, A=N7
i. f(x) = 4x+2, A=set of real numbers
ii. f(x) = 3+ 1/x, A=set of non zero real numbers
iii. f(x) = (2x+3) mod7, A=N7
Discrete Mathematics
Draw the Hasse diagram for the “divides” relation on {2, 3, 5, 10, 11, 15, 25, 36, 42, 108}
Discrete Mathematics
Determine whether each of these functions is a bijection from R to R?f (x) = (x2 + 1)/(x2 + 2)
