Get In-Depth Help with Logical Reasoning Solutions and Real-World Applications

Logical reasoning 97:89::59:____?

First order logic - why do we need function symbols?
Employing function symbols in first order logic necessitates us to delineate "terms" inductively, which renders many proofs more extensive and considerably more irksome.
Of course, function symbols simplify matters when trying to use first order logic to describe things, but on the surface it seems to me they could be replaced completely by relations: Instead of $f$ use a relation $R_f$ such that instead of writing $\mathrm{\phi <!--\; \phi \; -->}(f(x))$ write ($\mathrm{\forall <!--\; \forall \; -->}x\mathrm{\exists <!--\; \exists \; -->}!y(R(x,y))\wedge <!--\; \wedge \; -->R(x,y)\wedge <!--\; \wedge \; -->\mathrm{\phi <!--\; \phi \; -->}(y)$. Now use $f(x)$ as a shorthand notation, so you can use it in "real life" but avoid it in proofs.
I guess I'm missing some deep neccecity here, but what?
(The same goes for constant symbols, but they don't really complicate things as function symbols do).

Negation of "If ... then" statements
I am just being introduced to how logic is used in mathematics and my lecturer mentioned that $\sim <!--\; \sim \; -->(P\to <!--\; \to \; -->Q)\equiv <!--\; \equiv \; -->p\wedge <!--\; \wedge \; -->\sim <!--\; \sim \; -->q$. This is quite hard to grasp at first glance, so he gave an example: The negation of "if $x\ne <!--\; \ne \; -->0$ then $y=0$" is "$x\ne <!--\; \ne \; -->0\wedge <!--\; \wedge \; -->y\ne <!--\; \ne \; -->0$".
Well, my question is, why should that be the case? Why is the negation of "if $x\ne <!--\; \ne \; -->0$ then $y=0$" not "$x\ne <!--\; \ne \; -->0\wedge <!--\; \wedge \; -->y=0$"?
Any explanations will be greatly appreciated :)

Strengthening the Antecedent: From B implies C, infer $(A\wedge <!--\; \wedge \; -->B)$ implies C
How can I construct a Fitch style proof to prove this? I have tried
1. $B\to <!--\; \to \; -->C$
2. $A\wedge <!--\; \wedge \; -->B$
3. $B\wedge <!--\; \wedge \; -->Elim:2$
4. $C\to <!--\; \to \; -->Elim:1,3$
5. $(A\wedge <!--\; \wedge \; -->B)\to <!--\; \to \; -->C\to <!--\; \to \; -->Intro:2-4$

What does it mean when x is 'free'?
In the context of first-order logic and resolution (I'm trying to study Skolemization for a midterm tomorrow), I am seeing several references to $x$ being free or not free. What does this mean?
To pull all quantifiers in front of the formula and thus transform it into a prenex form, use the following equivalences, where $x$ is not free in Q:
Edit:
$\mathrm{\forall <!--\; \forall \; -->}xP(x)\wedge \mathrm{\exists <!--\; \exists \; -->}xQ(x)\equiv <!--\; \equiv \; -->\mathrm{\forall <!--\; \forall \; -->}x(P(x)\wedge \mathrm{\exists <!--\; \exists \; -->}xQ(x))\equiv \mathrm{\forall <!--\; \forall \; -->}x(P(x)\wedge \mathrm{\exists <!--\; \exists \; -->}yQ(y))\equiv <!--\; \equiv \; -->\mathrm{\forall <!--\; \forall \; -->}x\mathrm{\exists <!--\; \exists \; -->}y(P(x)\wedge Q(y)).$
If the second x is bound by the existential quantifier - why can it be pulled into prenex form if the blurb above says that this can be done only if x is not free? Or am I misunderstanding something?
Edit 2:
OK, i'm clearly not all here at the moment. Ignore that question

There are colored pencils in a box, 9 pencils of each color. It is known that there are a natural number of tens and a natural number of dozens of pencils in the box, while there are less than 300 pencils in the box. How many pencils are in the box?

Jacob guessed a natural number greater than 99 but less than 1000. The sum of the first and last digits of this number is 1, and the product of the first and second digits is 7. What number did Jacob guess?

As of 2015, there were 152 public nature reserves and national parks in the United States. How many nature reserves in the US and how many national parks, if there are 58 more reserves than parks?

After traveling 387 miles from Los Angeles to San Francisco, Rick noted that his car's odometer reads the same backwards as forwards. What is the next such number and how far will he have to drive to get it to appear on the odometer? Wirte and equation based on the facts of the problem.

A manufacturing firm finds that 70% of its new hires turn out to be good workers and 30% become poor workers. All current workers are given a reasoning test. Of the good workers, 60% pass it; 30% of the poor workers pass it. Assume that these figures will hold true in the future. If the company makes the test part of its hiring procedure and only hires people who meet the previous requirements and also pass the test, what percent of the new hires will turn out to be good workers?
The percent of new hires that will be good workers is %.

Write its converse. If the converse also true, combine the statements as a biconditional
If p $\to <!--\; \to \; -->$ q is true, then $\sim <!--\; \sim \; -->$ q $\to <!--\; \to \; -->$$\sim <!--\; \sim \; -->$ p is true

Let a1 a2 a3 .....an be geometric sequence find each indicated quantities a1= 100 a6 =1 r=? please show me how

A company has introduced a process improvement that reducesprocessing time for each unit, so that output is increased by 25%with less material, but one additional worker required. Underold process, five workers could produce 60 units perhour. Labor costs are $12/hour, and material input waspreviously $16/unit. For the new process, material isnow$10/unit.
Overhead is charged at 1.6 times direct laborcost. Finished units are sold for $31 each. What increase(or decrease) in productivity is associated with the process improvement?
(a) Productivity with old system
(b) Productivity with new system
(c) % change (increase or decrease) in productivity

Write its converse. If the converse also true, combine the statements as a biconditional
Algebra if x = 3, then $\mid <!--\; \mid \; -->x\mid <!--\; \mid \; -->$ = 3

Write its converse. If the converse also true, combine the statements as a biconditional
In the United States, if it is July 4, then it is Independence

Write its converse. If the converse also true, combine the statements as a biconditional
If a number is divisible by 20, then it is even

Compute working hours
in 7:30 out 11:50
in 12:15 out 17:00

Write its converse. If the converse also true, combine the statements as a biconditional
Algebra if x = 12, then 2x - 5 = 19

Find the elapsed time from 6:15 a.m. to 2:30 p.m.

Marco came to a drawbridge that had a sign stating that the tolltaker would double his money each time he crossed the bridge, but Marco would have to paya fee of $1.20 per crossing. Marco decided he would cross the bridge 3 times, tosee how the system worked.
Sure enough, each time Marco crossed the bridge the toll takerdoubled the amount of money in Marco's pocket and then took his $1.20 toll charge. Marcowas surprised when , after the third crossing, he paid the $1.20fee and found himself with no money.