Solution to PREDICATE LOGIC.(25 pts) A. Let P(x) be the statement x 2 > x4. If the … - Sikademy
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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

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Here's the Solution to this Question

(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.

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Question ID: mtid-5-stid-8-sqid-3167-qpid-1866