Which of the following sentences are statements? Give reasons for your answer.
i) Are the buses running today?
ii) What a pleasant weather!
iii) x2 +1 = 0 for a real value of x.
iv) Every odd number is a prime.
In the context of discrete mathematics, a statement is a sentence that can be classified as either true or false, but not both. Let's analyze each of the given sentences:
i) "Are the buses running today?" - This is not a statement. It is a question and does not express a definitive truth value of true or false.
ii) "What a pleasant weather!" - This is not a statement. It is an exclamation and does not make a
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Express 3x4+2x3 - 2x2+x in terms of [x]4, [x]3, [x]2 and [x].
Make a table of the values of the Boolean function
f (x1;x2;x3) = x2(x1
x3)
Write the function in DNF using the table.
Let t(n) denote the number of ways in which 2n tennis players can be paired to play n matches. Determine a recurrence equation for the sequence t(n).
Which of the following sentences are statements? Give reasons for your answer.
i) Are the buses running today?
ii) What a pleasant weather!
iii) x2 +1 = 0 for a real value of x.
iv) Every odd number is a prime.
Show that any tree with exactly two vertices of degree 1 is a path.
Which of the following statements are true and which are false? Give reasons for your answer.
i) ( ~p v q) and (p q) are logically equivalent.
ii) (m+n)! > m!+n! for m, n any positive integers.
iii) The number of distributions of m indistinguishable objects into n distinguishable containers is the same as the number of non-negative integral solutions of the equation x1+x2+....+xn = m.
iv) Sn, the number of subsets of a set with n elements, satisfies a second order linear homogeneous recurrence equation with constant coefficients.
v) K3,4 is Hamiltonian.
If an apple costsRs.15, a papaya costs Rs.17 and a banana costs Rs.8, write the generating function for the number of ways of purchasing these fruits with Rs.n.