Discrete Mathematics
Q201.
Geetha has a conjecture about integers, which is of the form\forall x\left [P(x)\Rightarrow \exists yQ(x,y) \right ] where P is a statement about integers, and Q is a statement about pairs of integers. Which of the following (one or more) option(s) would imply Geetha's conjecture?Q202.
Consider the first-order logic sentence \varphi \equiv \exists s\exists t\exists u\forall v\forall w\forall x\forall y\varphi (s,t,u,v,w,x,y) where \varphi (s,t,u,v,w,x,y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose \varphi has a model with a universe containing 7 elements. Which one of the following statements is necessarily true?Q203.
The statement (\neg p)\Rightarrow (\neg q) is logically equivalent to which of the statements below? I. p\Rightarrow q II. q \Rightarrow p III. (\neg q)\vee p IV. (\neg p)\vee qQ206.
Consider the following recurrence: \begin{aligned} f(1)&=1; \\ f(2n)&=2f(n)-1, & \text{for }n \geq 1; \\ f(2n+1)&=2f(n)+1, & \text{for }n \geq 1. \end{aligned}Then, which of the following statements is/are TRUE?MSQQ207.
Let p,q, r, s represent the following propositions. p: x\in{8,9,10,11,12} q: x is a composite number r: x is a perfect square s: x is a prime number The integer x\geq2 which satisfies \neg((p\Rightarrow q)\wedge (\neg r \vee \neg s)) is ________.Q208.
Which one of the following is NOT logically equivalent to \neg \exists x(\forall y(\alpha )\wedge \forall z(\beta ))?Q209.
Given thatB(a) means "a is a bear"F(a) means "a is a fish" andE(a,b) means "a eats b"Then what is the best meaning of\forall x[F(x) \rightarrow \forall y(E(y, x) \rightarrow b(y))]Q210.
Let H_1, H_2, H_3, ... be harmonic numbers. Then, for n \in Z^+, \sum_{j=1}^{n} H_j can be expressed as