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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
Discrete Mathematics
Use a proof by contradiction to show that there is no rational number r for which r+r+1=0.
(Assume that r=a/h is a root, where a and b are integers and a/b is in lowest terms.obtain an equation involving integers by multiplying by then look at whether a and bare each odd or
even.
(Assume that r=a/h is a root, where a and b are integers and a/b is in lowest terms.obtain an equation involving integers by multiplying by then look at whether a and bare each odd or
even.
Discrete Mathematics
Q#: A binary operation ∇ on [0,1] is a t-norm if and only if x∇(y∇z)=(x∇y)∇z
Discrete Mathematics
Determine whether each of these functions is a bijection from R to R.
f (x) = (x2 + 1)/(x2 + 2)
f (x) = (x2 + 1)/(x2 + 2)
Discrete Mathematics
Create the equivalent logic circuit of the following logic expression:
1. Q = (A + B) . (C +D)'
2. F1 = (A + BC') . D
3. Q = [(A + B)' . C] +B . C
Discrete Mathematics
Find the complement of the following expression using dual of a function:
1. xy' + x'y
2. (AB' + C) D' + E
3. AB (C'D + CD') + A'B' (C' +D) (C + D')
4. (x + y+ z) (x' + z') (x + y)
Discrete Mathematics
Boolean expressions to minimal number of literals:
1. x'y' + xy + x'y
2. (x + y) (x + y')
3. x'y + xy' + xy + x'y'
4. x' + xy +xz' + xy'z'
5. A'C' + ABC + AC'
6. (x'y' + z)' + z + xy + wz
7. A'B (D' + C'D) + B (A + A'CD)
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.
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#340153 on Dec 2023
#340153 on Dec 2023