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SY Common ISE ODD 25-26

SY Common ISE ODD 25-26

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Express uniqueness: every positive integer n can be written as k*2^m with k odd, uniquely. Which logical form captures uniqueness?

∃k∃m (k odd ∧ n=k*2^m ∧ ∀k'∀m'(n=k'*2^{m'} → k'=k ∧ m'=m))

∀k∃m (k odd ∧ n=k*2^m ∧ ∀k'∀m'(n=k'*2^{m'} → k'=k ∧ m'=m))

∃k∀m (k odd ∧ n=k*2^m ∧ ∀k'∀m'(n=k'*2^{m'} → k'=k ∧ m'=m))

∃k∃m (n=k*2^m)

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The reflexive-transitive closure of R on {1,2,3} with R = {(1,2),(2,3)} has size:

Transform the statement ¬(∀x ∃y (P(x) → Q(y))) into an equivalent form with all negations pushed inside

∃x ∀y (P(x) ∧ ¬Q(y))

∀x ∃y (¬P(x) ∧ Q(y))

∃y ∀x (¬P(x) ∧ ¬Q(y))

∃x ∀y ¬(P(x) → Q(y))

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n+1

2n+1

n-1

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In a directed graph, the entries of the incidence matrix can be:

1 and 2

0, 1, and –1

Only 0 and –1

Only 0 or 1

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Which of the following is not a partition of {1,2,3,4}?

{{1,2}, {2,3,4}}

{{1,2,3}, {4}}

{{1,2}, {3,4}}

{{1}, {2}, {3}, {4}}

What is the number of edges present in a wheel graph W_n?

n+1

n/2

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P and Q are true

P is false and Q is true

P and Q are false

P is true and Q is false

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Total number of edges of a complete graph Km,n

mn/2

m + n

m − n

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If 10 keys are hashed into 7 slots, by Pigeonhole Principle, at least one slot contains ≥ ? keys.

❌

✅

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