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MAT2051 DISCRETE MATHEMATICS EXAM Q & A WITH RATIONALES 2024.
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MAT2051 DISCRETE MATHEMATICS EXAM Q & A WITH RATIONALES 2024.
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MAT2051Discrete Mathematics
LATEST EXAM w/ RATIONALES
2024
,PART A:
1. Consider the following proposition: "For all natural
numbers n, if n is even, then n + 1 is odd." Which of the
following is a valid way to prove this proposition by
contradiction?
a) Assume that there exists a natural number n such that n is
even and n + 1 is even, and derive a contradiction.
b) Assume that there exists a natural number n such that n is
odd and n + 1 is odd, and derive a contradiction.
c) Assume that for all natural numbers n, n is even and n + 1
is even, and derive a contradiction.
d) Assume that for all natural numbers n, n is odd and n + 1
is odd, and derive a contradiction.
*Answer: a) This is the correct way to prove the proposition
by contradiction. If we assume that there exists a natural
number n such that n is even and n + 1 is even, then we can
write n = 2k and n + 1 = 2l for some natural numbers k and
l. But then we have 2l - 2k = 1, which implies that 1 is
divisible by 2, which is a contradiction.*
2. Consider the following relation R on the set {a, b, c, d}:
R = {(a, a), (a, b), (b, b), (b, c), (c, c), (c, d), (d, d)}. Which
of the following statements are true about R?
a) R is reflexive
b) R is symmetric
c) R is transitive
d) R is antisymmetric
*Answer: a) and c) R is reflexive because for every element
, x in the set, (x, x) belongs to R. R is transitive because for
any elements x, y, z in the set, if (x, y) and (y, z) belong to
R, then (x, z) also belongs to R. R is not symmetric because
there are elements x and y in the set such that (x, y) belongs
to R but (y, x) does not belong to R. For example, (a, b)
belongs to R but (b, a) does not belong to R. R is not
antisymmetric because there are elements x and y in the set
such that (x, y) and (y, x) belong to R but x is not equal to y.
For example, (b, c) and (c, b) belong to R but b is not equal
to c.*
3. Consider the following function f: N -> N defined by f(n)
= 3n + 2 for every natural number n. Which of the following
statements are true about f?
a) f is one-to-one
b) f is onto
c) f has an inverse function
d) f is bijective
*Answer: a) and c) f is one-to-one because for any natural
numbers m and n, if f(m) = f(n), then we have 3m + 2 = 3n
+ 2, which implies that m = n. f is not onto because there are
natural numbers that are not in the range of f. For example,
1 is not in the range of f because there is no natural number
n such that f(n) = 1. f has an inverse function because it is
one-to-one. The inverse function of f is g: N -> N defined
by g(n) = (n - 2)/3 for every natural number n such that n - 2
is divisible by 3. f is not bijective because it is not onto.*
4. Consider the following algorithm that takes as input a
positive integer n and returns as output a positive integer:
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