Axiom Math-1. If S is a student required to take Calculus II, then S is either a

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Axiom Math-1. If S is a student required to take Calculus II, then S is either a mathematics major or a chemistry major.

Axiom Math-2. If S is a student required to take Calculus I, then S is required to take Calculus II or is required to take Statistics.

Axiom Math-3. S is a student required to take Statistics if and only if S is neither a mathematics major nor a chemistry major.

Axiom Math-4. Every student is either required to take Statistics or required to take Calculus I.

Axiom Math-5. There is a student that is both a mathematics major and a chemistry major. Axiom Math-6. There is a student that is both required to take Calculus I and required to take Statistics.

Theorem 0.4. If S is a mathematics major, then S is required to take Calculus I. Here we have to establish that this implication is true no matter who S is. The way that we do this is by saying in our proof that S is arbitrary. This means that our argument is a hypothetical one that can apply to any student. Consider the following proof of the theorem above. Proof. Suppose that S is an arbitrary mathematics major. By Axiom Math-3, S is not required to take statistics. Hence by Axiom Math-4, S is required to take Calculus I. As S represented an arbitrary student, we have proved our implication. Note that our proof made no assumption about S other than that S is a mathematics major (as this was the hypothesis). So, for any possible person that we could substitute for S, either that person is not a mathematics major (and hence the implication is true for him/her), or the argument demonstrates that the person is required to take Calculus I. So we have established that the implication is true for all values of S. A slightly different type of proof is when we show that something must exist. Consider the following theorem.

Theorem 0.5. There is a student that is required to take Calculus II. The proof of this theorem will not start out with “let S be an arbitrary student”. That is because we do not wish to show that every student is required to take Calculus II (indeed, this is false). Instead, we only need to show that there is at least one student that is required to take Calculus II. How do we do this when we do not know the names of any of the students? While we do not know the names of students, we do have some existence axioms that identify some particular students for us. So, our proof will start out by using one of these axioms and giving that student a label. Proof. By Axiom Math-5, there is a student that is both a mathematics major and a chemistry major; call this student A. By Axiom Math-3, A is not required to take statistics. So, by Axiom Math-4, A is required to take Calculus I. Therefore, by Axiom Math-2, A is either required to take Calculus II or required to take statistics. However, we already know that A is not required to take statistics, A is required take Calculus II. Note that in the proof above, A is not arbitrary. We may not use it to establish an implication about all mathematics and chemistry majors. Instead, we may only prove specific facts about that student A. Now it is your turn again: Problem 0.4. Prove that if S is a student required to take Calculus II, then S is also required to take Calculus I. Problem 0.5. Prove that there is a student that is not required to take Calculus II

Problem 0.4. Prove that if S is a student required to take Calculus II, then S is also required to take Calculus I.

Problem 0.5. Prove that there is a student that is not required to take Calculus II

Explanation / Answer

Prove that if S is a student required to take Calculus II, then S is also required to take Calculus I

given S is a student required to take calculas II then by axiom Math1 -S is either mathematics major or chemistry major

and by axiom Math 3 S is the student that is not required to take statistics

by axiom Math 4- student S is required to take Calculas I as S is not from statistics

hence S is also required to take Calculas I.

(0.5)    Prove that there is a student that is not required to take Calculus II

Suppose if possible there is no student that is not required to take Calculas II

that is every student is required to take Calculas II BUT THIS IS NOT POSSIBLE For evry student there can be a student S WHICH IS required to take Calculas I and statistics by axiom Maths6 and this student S is not required to take Calculas II

HENCE there is a student that is not required to take Calculus II.

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