PREDICATE LOGIC. Rewrite each sentence symbolically and determine the truth values. Write T if it is true and F if it is false. Show complete solution. (5 pts each) 1. For some integer x, 𝑥 = 𝑥2 − 2 2. For every real number x, 𝑖𝑓 𝑥2 − 1 > 𝑥 𝑡ℎ𝑒𝑛 𝑥 + 1 > 1 3. For some integer n, 4n = 3n + 1 F. RULE OF INFERENCE. Determine if the following argument is valid. If it is valid, what rule of inference is used in each of the following arguments? Show solution. (4 pts each) 1. Joy wrote a C++ source code, or Jen wrote a Java source code. If Joy wrote a C++ source code, then the problem was solved. If Jen wrote a Java source code, then the problem was solved. 2. There does not exist someone who likes to be COVID – 19 positive; hence, everyone does not like to be vaccinated.

E.

1.Given, $x=x^2-2$

at $x=2, LHS=x=2$

$RHS=x^2-2=(2)^2-2=4-2=2=LHS$

Hence The given statement is true and Truth value is T for some integer $x.$

2.if $x^2-1>x,$ then $(x+1)>1$

At $x=2, x^2-1i.e. (2)^2-1=3>2$

Then, ($x+1)i.e. (3)>1$ ,Which is true

The given statement is true with truth value T for every real number $x$ .

3.Given, $4n=3n+1$

at $n=1, LHS=4(1)=4$

$RHS=3(1)+1=4=LHS$

So Given statement is true and Truth value is T for some integer $x$ .

F.

1.If Joy write c++ code then Problem was solved. Also If jen write a java code, Then problem was also solved. The Given statemnt is Valid with Rule of inference is Contrapositive reasoning.

2.There does not exist someone who likes to be COVID – 19 positive, But everyone needs to be vaccinated, as we don't know Whether corona might comes in future. So Given statement is invalid

with Rule of inference used is Fallacy of converse.