The statement “An integer is even if and only if its square is even”

1. (5 points) True/False Let S = {a, b, c}
(a) T F a ∈ P(S)
(b) T F {a} ∈ P(S)
(c) T F {a, b} ⊆ P(S)
(d) T F {a, {a, b}} ⊆ P(S)
(e) T F {{a}, {a, b}} ⊆ P(S)
2. (10 points) Consider the statement “An integer is even if and only if its square is even”
(a) Translate the statement into quantified logic. Be sure to specify a universe, and define
any open statements you use.
(b) Prove the statement.
3. (10 points) For any sets A and B: A ∩ B = A ∪ B
(a) Sketch two Venn Diagrams to convince yourself (and me) of this property.
(b) Now prove this property using Laws of Logic.
4. (5 points) Recall the Fibonacci Sequence:
F0 = 0, F1 = 1, Fn = Fn−1 + Fn−2
Prove by induction:
Xn
i=0
F2i = F2n+1 − 1
5. (5 points) Use the division algorithm to express the number 500 in base 7.
6. (10 points) Let a = 232 and b = 144.
(a) Use The Euclidean Algorithm to find gcd(a, b).
(b) Find integers x and y so that: gcd(a, b) = ax + by
7. (10 points) Construct counter-examples to the following implications:
(a) If A ∪ C = B ∪ C then A = B
(b) If A ∩ C = B ∩ C then A = B
8. (5 points) Prove: For all a, b, c ∈ Z, [(a|b) ∧ (b|c))] ⇒ a|c.
9. (5 points) Prove: For all a, b ∈ Z, [(a|b) ∧ (b|a))] ⇒ a = ±b.

Socrates and Plato utilized the Allegory Cave to clarify the basic considering aptitudes normal society by clarifying what the envisioned the truth should resemble and what the genuine truth resembles this present reality outside the cavern.
They accepted life resembled being in a cavern and that is the thing that life was about and seeing shadows escaped reality. These logicians both accepted that this relationship of the individuals in the cavern not understanding the man who informed them regarding the outside life and there’s living things and things you can contact and believe, and the shadows they were seeing were individuals strolling by with creatures and crops and those were living individuals.

A few people characterize bliss as something that is “noticeable and substantial, for example, joy, riches or respect. What’s more, it varies from individual to individual, yet it can likewise mean various things to a similar individual at various occasions. For instance, on the off chance that I am wiped out, wellbeing is what I’m making progress toward. On the off chance that I am poor, riches is what I’m making progress toward. Hence, we comprehend that there’s an extreme decent on the grounds that there is a reason for goodness in them. We have a natural inclination that the great is something that has a place with us and that it isn’t something that can be handily detracted from us.

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Kant accepted profound quality to be deontological; the word ‘Deon’ when interpreted is the Greek expression for obligation. Consequently the deontological contention depends on doing acts since they are acceptable in themself and not founded on the outcomes or different components including feeling. Kant guessed that one had the option to make an ethical framework dependent on reason. People have reason above impulse, and this implies our inspirations go past negligible delight .In this article I will clarify and talk about the deontological contention supported by Kant.

Kant addresses being acceptable and presumes that positive attitude is the will to make the best decision. It’s bad a direct result of what you can receive in return but since it is acceptable in itself. People are flawed and will in general act as per their wants, which isn’t ethically acceptable. A model being if a businessperson gave their client the right measure of progress; on the off chance that they did this because of dread of being gotten out by giving their client an inappropriate the change then the demonstration is certainly not a by and large great act be that as it may on the off chance that they restored the right change not considering the outcomes however doing it for the correct explanation and simply keeping in mind the ethical standards. Ethical quality is an arrangement of decides that you place on yourself; moral principles come to us, as we are balanced creatures. A few reasons we can’t overlook and apply to everybody.

Clear cut objectives are sayings; this being decides that you need to do constantly paying little mind to conditions. Kant gives three distinct plans about the straight out goals. The first of the three plans is universalisation he composes ‘ I should never to act aside from so that I could likewise will that my saying ought to become universalised’. By saying this he expresses that one should just act such that bodes well for you yet in addition in a way that is significant and would permit every other person to act similarly. An individual’s ethical law is their will and should be predictable. A model being one must not lie, this is a significant proverb as though everybody lied the world would be degenerate and genuineness would be a futile attribute as falsehoods would be the standard and reality would be lost in interpretation. It very well may be accepted that our activities are simply us looking for joy and maintaining a strategic distance from torment anyway Kant presents an unmistakable clarification in ‘Preparation of a Metaphysic of Moral’ where he contends against this point by expressing that in the event that joy were the main thing to spur our activities, at that point we would be just have nature to direct us, as intuition gets the job done for getting delight.

The second of the three details is to ‘act so that you treat mankind, regardless of whether it is in your own individual or in the individual of another, consistently as an end, and never just as a methods’ this passes on that one ought to consistently regard different people groups status as creatures in the ethical worth and regard ourselves. Its justifiable a model being cab drivers, medical caretakers and baristas are the way to the finish of whatever administrations they’re giving anyway this is permitted as the administration is given willfully and not in a manner which eventually powers or removes their unrestrained choice. Along these lines in rundown of the subsequent definition, this is to not utilize individuals as intends to and end.

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