Question

PREDICATE LOGIC.(25 pts)

A. Let P(x) be the statement x 2 > x4. If the domain consists of the integers,

what are the truth values?

1. P(0)

2. P(-1)

3. P(1)

4. P(2)

5. ∃xP(x)

6. ∀xP(x)

B. Write the following predicates symbolically and determine its truth value.

Note: Use at least three (3) values for the variables. (5 pts each)

1. for every real number x, if x>1 then x – 1 > 1

2. for some real number x, x2 ≤ 0

C. Translate the following English sentence into symbol. (3 pts each)

1. No one in this class is wearing pants and a guitarist.

Let:

Domain of x is all persons

A(x): x is wearing pants

B(x): x is a guitarist

C(x): belongs to the class

2. No one in this class is wearing pants and a guitarist.

Let:

Domain of x is persons in this class

A(x): x is wearing pants

B(x): x is a guitarist

3. There is a student at your school who knows C++ but who doesn’t

know Java.

Let:

Domain: all students at your school

C(x): x knows C++

J(x): x knows Java

Answer 1

(A) To answer these questions, we first solve the indicated inequality $\left(x^2>x^4\right)$ for all real numbers $x\in\mathbb{R}$ .

$$x^2-x^4>0\to\left.x^2\cdot(1-x^2)>0\right|\cdot(-1)\to\\[0.3cm] x^2\cdot\left(x^2-1\right)\equiv\boxed{x^2\cdot(x-1)(x+1)<0}$$

Conclusion,

$$\boxed{x\in(-1;0)\cup(0;1)}\\[0.3cm] \text{As you can see, the solutions of this inequality are NOT integers}$$

Moving on to the answers to these questions :

1. $P(0) : 0^2>0^4-\boxed{\text{FALSE}}$

2. $P(-1) : (-1)^2=1>(-1)^4=1-\boxed{\text{FALSE}}$

3. $P(1) : 1^2=1>1^4=1-\boxed{\text{FALSE}}$

4. $P(2) : 2^2=4>2^4=16-\boxed{\text{FALSE}}$

5. $\exists xP(x) : x^2>x^4-\boxed{\text{FALSE}}$

6. $\forall xP(x) : x^2 >x^4-\boxed{\text{FALSE}}$

(B) Let $P(x)$ be the statement $(x>1)$ and $Q(x)$ is $\left(x-1>1\right)$. Then, the sentence " for every real number x, if $x>1$ then $x-1>1$ " has the form

$$\boxed{\forall x\in\mathbb{R}\left(P(x)\to Q(x)\right)-\text{FALSE}}$$

For example,

$$x=1.1\to\left\{\begin{array}{l} P(1.1)=1.1>1-\text{TRUE}\\ Q(1.1)=1.1-1=0.1>1-\text{FALSE} \end{array}\right. \\[0.3cm] x=1.2\to\left\{\begin{array}{l} P(1.2)=1.2>1-\text{TRUE}\\ Q(1.2)=1.2-1=0.2>1-\text{FALSE} \end{array}\right. \\[0.3cm] x=1.3\to\left\{\begin{array}{l} P(1.3)=1.3>1-\text{TRUE}\\ Q(1.3)=1.3-1=0.3>1-\text{FALSE} \end{array}\right. \\[0.3cm]$$

Let $P(x)$ be the statement $(x^2\le0)$. Then, the sentence " for some real number $x$ , $x^2\le0$ " has the form

$$\boxed{\exists xP(x)-\text{TRUE}}$$

For example,

$$x=0\to 0^2\le0-\text{TRUE} \\[0.3cm] x=1.2\to 1.2^2\equiv1.44\le0-\text{FALSE}\\[0.3cm] x=-2.5\to(-2.5)^2\equiv6.25\le0-\text{FALSE}$$

(C)

1.No one in this class is wearing pants and a guitarist.

Let:

Domain of $x$ is all persons

$A(x) : x$ is wearing pants

$B(x) : x$ is a guitarist

$C(x) :$ belongs to the class

$$\boxed{\forall x\in D,\left(C(x)\to\left(A(x)\land B(x)\right)\right)}$$

2. No one in this class is wearing pants and a guitarist.

Let:

Domain of $x$ is persons in this class

$A(x):x$ is wearing pants

$B(x):x$ is a guitarist

$$\boxed{\forall x\left(\neg A(x)\land\neg B(x)\right)}$$

3. There is a student at your school who knows C++ but who doesn’t

know Java.

Let:

Domain: all students at your school

$C(x):x$ knows C++

$J(x):x$ knows Java

$$\boxed{\exists x\left(C(x)\land\neg J(x)\right)}$$

Q.E.D.