Question
Check whether the following statements are true or not. Justify your answers with a short proof or a counter example.
i) If the contrapositive of a statement is true, then the statement itself is also true.
ii) an+3an-1+2an-2= 2" is a linear homogeneous recurrence relation.
iii) A particular solution of the recurrence relation an-2an-1 + an-2=1 has the form Cn2.
iv) There exists a boolean expression in variables x,x, and x, with CNF as
v) If a dice is rolled thrice, then the probability of getting a 6 each time is
vi) Every odd cycle has the same chromatic and edge chromatic numbers.
vii) Every Eulerian graph is Hamiltonian.
viii) gives the number of ways in which any 3 objects can be placed in any 4 boxes.
ix) There exists a self-complementary planar graph on 5 or more vertices.
x) The number of partitions of 6 is 10.
Answer :
Word Count : 896
Let's go through each statement and determine whether it is true or false, justifying the answers with a short proof or a counterexample. --- ### (i) If the contrapositive of a statement is true, then the statement itself is also true. - A statement of the form \( P \Rightarrow Q \) has a contrapositive \( \neg Q \Rightarrow \neg P \). - The contrapositive of a true statement is always true, and vice versa. - Verdict: True ✅ --- ### (ii) \( a_n + 3a_{n-1} + 2a_{n-2} = 2 \) is a linear homogeneous recurrence relation. - A linear homogeneous recurrence relation has the form: \[ c_k a_n + c_{k-1} a_{n-1} + \dots + c_1 a_1 + c_0 a_0 = 0 \] - Here, the right-hand side is not zero (\(2 \neq 0\)), so it is not homogeneous. - Verdict: False ❌ --- ### (iii) A particular solution of the recurrence relation \( a_n - 2a_{n-1} + a_{n-2} = 1 \) has the form \( Cn^2 \). - The corresponding homogeneous recurrence is \( a_n - 2a_{n-1} + __________ ______ _______ __________ _______ ___ _________ _________ __________.
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Let's go through each statement and determine whether it is true or false, justifying the answers with a short proof or a counterexample. --- ### (i) If the contrapositive of a statement is true, then the statement itself is also true. - A statement of the form \( P \Rightarrow Q \) has a contrapositive \( \neg Q \Rightarrow \neg P \). - The contrapositive of a true statement is always true, and vice versa. - Verdict: True ✅ --- ### (ii) \( a_n + 3a_{n-1} + 2a_{n-2} = 2 \) is a linear homogeneous recurrence relation. - A linear homogeneous recurrence relation has the form: \[ c_k a_n + c_{k-1} a_{n-1} + \dots + c_1 a_1 + c_0 a_0 = 0 \] - Here, the right-hand side is not zero (\(2 \neq 0\)), so it is not homogeneous. - Verdict: False ❌ --- ### (iii) A particular solution of the recurrence relation \( a_n - 2a_{n-1} + a_{n-2} = 1 \) has the form \( Cn^2 \). - The corresponding homogeneous recurrence is \( a_n - 2a_{n-1} + __________ ______ _______ __________ _______ ___ _________ _________ __________.
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Related Questions
For any statements p,q and r, prove that
समुच्चय से (1,2,3,4) पर सभी आच्छादक फसनी की सूची बनाइए। समुच्चय
से (1,2,3,4,5) पर किन आउादक फलन हैं?
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Solve the recurrence relation:
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सूरीय व्यंजक कार्क परिपथ बनाइए।
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