Define the following predicates: Real( x) = “x is a real number”. Pos(x) = “x is...

Define the following predicates:

Real( x) = “x is a real number”.

Pos(x) = “x is a positive real number.”

Neg(x) = “x is a negative real number.”

Int(x) = “x is an integer.”

Rewrite the following statements without using quantifiers. Determine which

statements are true or false and justify your answer as best you can.

a) Pos(0)

b) ∀x, Real(x) ∧ Neg(x) → Pos(−x)

c) ∀x, Int(x) → Real(x)

d) ∃x such that Real(x) ∧∼ Int(x)

Solutions

Expert Solution

Colby Messinger answered 3 months ago

Next > < Previous

Related Solutions

Prove or disprove the statements: (a) If x is a real number such that |x +...

Prove or disprove the statements: (a) If x is a real number such that |x + 2| + |x| ≤ 1, then x 2 + 2x − 1 ≤ 2. (b) If x is a real number such that |x + 2| + |x| ≤ 2, then x 2 + 2x − 1 ≤ 2. (c) If x is a real number such that |x + 2| + |x| ≤ 3, then x 2 + 2x − 1 ≤ 2....

Recall that the absolute value |x| for a real number x is defined as the function...

Recall that the absolute value |x| for a real number x is defined as the function |x| = x if x ≥ 0 or −x if x < 0. Prove that, for any real numbers x and y, we have (a) | − x| = |x|. (b) |xy| = |x||y|. (c) |x-1| = 1/|x|. Here we assume that x is not equal to 0. (d) −|x| ≤ x ≤ |x|.

“A real number is rational if and only if it has a periodic decimal expansion” Define...

“A real number is rational if and only if it has a periodic decimal expansion” Define the present usage of the word periodic and prove the statement

Suppose that x is real number. Prove that x+1/x =2 if and only if x=1. Prove...

Suppose that x is real number. Prove that x+1/x =2 if and only if x=1. Prove that there does not exist a smallest positive real number. Is the result still true if we replace ”real number” with ”integer”? Suppose that x is a real number. Use either proof by contrapositive or proof by contradiction to show that x3 + 5x = 0 implies that x = 0.

Define the random variable X to be the number of times in the month of June...

Define the random variable X to be the number of times in the month of June (which has 30 days) Susan wakes up before 6am a. X fits binomial distribution, X-B(n,p). What are the values of n and p? c. what is the probability that Susan wakes us up before 6 am 5 or fewer days in June? d. what is the probability that Susan wakes up before 6am more than 12 times?

2. Let x be a real number, and consider the deleted neighborhood N∗(x;ε). (a) Show that...

2. Let x be a real number, and consider the deleted neighborhood N∗(x;ε). (a) Show that every element of N∗(x;ε) is an interior point. (b) Determine the boundary of N∗(x;ε) and prove your answer is correct.

Let the following ODE be given. Let x be a real number. xy"+(1-x)y' = (k-1/2)y Give...

Let the following ODE be given. Let x be a real number. xy"+(1-x)y' = (k-1/2)y Give an explict solution for k=-2.5

We produce a random real number X through the following two-stage experiment. First roll a fair...

We produce a random real number X through the following two-stage experiment. First roll a fair die to get an outcome Y in the set {1, 2, . . . , 6}. Then, if Y = k, choose X uniformly from the interval (0, k]. Find the cumulative distribution function F(s) and the probability density function f(s) of X for 3 < s < 4.

Solve the system y'=x x'=-x+ry-y^3 r is a real number. Solve for eigenvalues & stability of...

Solve the system y'=x x'=-x+ry-y^3 r is a real number. Solve for eigenvalues & stability of system. Show bifurcation value.

Recall the following definition: For X a topological space, and for A ⊆ X, we define...

Recall the following definition: For X a topological space, and for A ⊆ X, we define the closure of A as cl(A) = ⋂{B ⊆ X : B is closed in X and A ⊆ B}. Let x ∈ X. Prove that x ∈ cl(A) if and only if every neighborhood of x contains a point from A. You may not use any definitions of cl(A) other than the one given.

Subjects