# Predicates and Quantifiers

a) ∀x(A → P(x)) ≡ A → ∀xP(x)

b) ∃x(A → P(x)) ≡ A → ∃xP(x)

My Solution

a) Suppose A is false.Then A -> P(x) is trivially true because if hypothesis is false then conditional statement is trivially true.

# I need help with this conditional probability question

Suppose P(S) = x, P(S') = 1-x then for Alice:

P(C) = P(C|S)P(S) + P(C|S')P(S') = 0.43

P(C) = P(C|S)P(S) + P(C|S')P(S') = 0.

# Discrete Mathematics - Proof Methods and Strategy

Suppose that $P(x)$ is the statement “$x2 +2x = 1$”.If the domain is the set of integers,
what are the truth values of $\forall x P(x)$ and $\exists x P(x)$?

# Existence and uniqueness of an abstract mathematical problem

Suppose P is an abstract mathematical problem, an element $x$ is a solution of P if P(x) is true.The uniqueness is defined as P is unique determined if any two solution of P is equal.

# If $P(a)=0 \Rightarrow P(a+1)=1$ then $P(x)$ has no repeated roots.

Let $P(x) \in \mathbb{R}[x]$ be polynomial with all real roots and has the property that $P(a)=0 \Rightarrow P(a+1)=1$ for all $a \in \mathbb{R}$.I think this problem statement is not true because if we suppose that $P(x)=x$ then $P(0)=0 \Rightarrow P(1)=1$, $\;P(x)$ has no repeated root.

# Equivalent definitions of perfect field.

Suppose that $T$ has characteristic $p>0,$ and that $x \mapsto x^p$ is not an automorphism.Since a ring homomorphism from a field into itself is injective or zero, then $x \mapsto x^p$ is not surjective.

# Proving a predicate does not imply another predicate.

Suppose we have arbitrary predicates are $P$ and $Q$.Consider:

$(x=0)$ as $P(x)$ and $(x > 0)$ as $Q(x)$.

# A question about Lp spaces

Suppose $f\in L^p[0,\infty)$ for some $p>0$.Show that $\lim_{x\to\infty}f(x)=0$.

# proof-explaination : $\forall x~(P (x ) ∧ Q(x )) \equiv \forall x~P (x )~ \wedge ~\forall x~Q(x )$

Content: suppose that $\forall x~(P(x) \wedge Q(x))$ is true.Because $P(a)$ is true and $Q(a)$ is true for every element in the domain, we can conclude that $\forall x~P(x)$ and $\forall x~Q(x)$ are both true.

# Integral inequality with $L^p$ norm

Suppose $f(x)$ is p-th integrable, i.Let $A_N = \{x\in A \, : \, |f(x)|>N\}$.

# Multiplying binomial distributions

Suppose I have X ∼ Bin(1,p).I know that X+X ∼ Bin(1+1,p), which would lead me to the (probably incorrect) assertion that X^2 ∼ Bin(1*1,p).

# If $p(x,y)$ is a homogeneous polynomial, and $p(x,\lambda x)$ is identically $0$, then $y-\lambda x$ is a factor of $p(x,y)$

Suppose $p(x,y)$ is a homogeneous polynomial over $\mathbb{C}$, say, and
$p(x,\lambda x) = 0$ for some $\lambda \in \mathbb{C}.$ Then why exactly must $y- \lambda x$ divide $p(x,y)$?

# Finding the proabability P(Y=30)

Suppose that , X is chosen uniformly from {1,2,3,.What is P(Y=30)?

# Probability conditioning- which interpretation is correct?

Suppose $A$ and $X$ are random variables and suppose we have $VaR_u(AX)=5$.Furthermore, suppose we are interested in the following probability $P(AX>VaR_u(AX)|A=a)$.

# Moment generating function of multinomial distribution

Suppose that $X$ is a Multinomial($n, \textbf{p}$) r.v.