Social: Don't mistake the perception of anonymous communication on the Internet as a license to hurt other peoples feelings. Some of those people are crazy and you are probably not as anonymous as you think. Don't get into "Is so/Is not" discussions. Arm yourself with reliably sourced facts and compelling arguments for or against a specific position. Then ask yourself if jumping into the conversation is a worthy use of your time. Time something that is easy to spend and easy regret wasting. Don't trust people who offer you free online services or free credit cards. Once you give out your email address/personal details you can't control what use these people will have for them. People will call you claiming to be fixing things and ask for your credit card, account numbers or password over the phone. Don't engage. Simply hang up on these probable criminals and research their claims independently. Keep your communication lines open to your family and friends. These social contacts will be your future lifelines -- and not just in college. Financial Make a spending budget based on what you know and what you guess. Use broad categories for entertainment. Track your budget. Adjust your budget to fit reality. Use the budget to plan for the future. Balance your checkbook regularly against checks and debit-card charges. Make sure these numbers make sense with respect to the money the bank says you have and the budget. Don't skimp on educational materials. If a course lists books as optional, buy them. Don't surround yourself with trivia. Regularly clean up your workspace and resell/give-away your non-academic books to make more room. Educational Find a "quiet" place to think and study which may involve putting on your headphones and playing music. TV and the Internet can be harder to ignore when you are working on difficult questions. Don't rely on the lecturer/professor to explain the material to you. Before the lecture, read the textbook chapter(s) and work through as many exercises as you can. That way, you will have background for what the lecturer will be going over and you will have context for any jokes. Especially in large classes devoted to core materials, there may not be time for exploring all aspects of a topic. So be cautious about derailing the lecture plan by asking an out-of-sequence question. Different lecturers will prefer questions in the middle of the material, some set aside time at the end of the hour, and some prefer questions only at their office hours. In the sciences, don't just do calculations and write down your answer. Document your thinking so that even if you don't know how to do all the steps, you get partial credit for going about solving the problem the right way. Also, showing your work helps if your calculator computes the wrong type of logarithm or uses the wrong type of angular measure at some point. When you have time, tackle the book problems that weren't assigned to you. Extra practice with the material can be very useful when it comes to examination time. Computers (General) Don't write down your passwords where they can be stolen. Don't clutter up the top level of your home directory. Create projects directory with a new directory for every class and and new subdirectory for every projects. Computers (Macintosh-centric) Learn to do things in Terminal windows. Terminal is an application that you should add to your Dock because it gives you access to the Unix-flavor command line interface behind all modern Macintosh computers. Macintosh computers come with Time Machine backup software pre-installed. Consider buying a 500 GB - 2 TB external drive exclusively for Time Machine to automatically backup to. (Never physically disconnect an external drive until the operating system has finished using it. On the Macintosh this is done by "ejecting" the drive from a Finder window or "dragging the drive icon to the trash.") Macintosh computers come with perl and python pre-installed and these scripting languages can be used to do many things. To get a full-fledged development environment there is a large (GB) free download called Xcode from the Apple Store that provides C compilers and more. Don't download this over dialup. Even though Macintosh computers come with programming tools, you might get programming environment closer to the C standard programming environment if you use Linux. If you follow the instructions on https://www.vagrantup.com/ you can download two programs that will make Linux/Unix/C development much easier because a "virtual" Linux box can be set up in any subdirectory, sharing disk access, CPU and memory with your real machine. Commands in the Terminal window like "vagrant up" and "vagrant ssh" give you access.
Math Backgrounder Mathematics, as an occupation, is the process of building castles of logic. The foundation of each castle is a statement of axioms and definitions which are accepted as universally true within the domain. From that starting point, the truth of various statements is shown to be proven from the axioms, and each newly proven theorem allows the castle to be built higher and higher. If ever a statement is proven to be both false and true, then that is a contradiction and causes the castle to collapse. A contradiction shows that not all of the starting assumptions can be true at once. Mathematics, as a body of knowledge, is a catalogue of all the castles and how they relate to each other. Logic Formal logic is about what makes a rigorous argument — something without room to quibble or hedge. Thus, it is an abstraction of real life arguments about what we know only imperfectly. The strongest form of mathematical proof is a formal proof in logic, but in practice those tend to be long and dry and even in academic journals usually only short informal proofs are given with the reader expected to bridge the gaps left formally open with his own background knowledge. Because logic is formal, it has it’s own peculiar language of symbols which are used to make sure that what is conveyed is no more and no less than what is meant to be conveyed. These are some examples of the symbols used in propositional logic, the logic of sentences which can be true or false. * Logical consequence, Logical entailment: A,B,C,D,E,…,Y ⊨ Z or A,B,C,D,E,…,Y ⇒ Z One sentence (Z) is said to be a logical consequence of a set of sentences, if and only if, in virtue of logic alone, it is impossible for the sentences in the set (A…Y) to be all true without the other sentence (Z) being true as well. This is the glue that holds together mathematics. A formal proof shows all the steps from (A…Y) to Z and explains exactly why each step is valid. * Material conditional, material implication, material consequence: … ⊨ A → B, if A then B, B only if A This is the sentence that it is never that case that B is false and A is true. A doesn’t necessarily cause B to be true. * True: ⊤, T The simplest sentence we could assert. * False: ⊥, F The simplest sentence we could never assert. * Logical Negation: !A, ¬A How we assert something is not true. * Modus Ponens A rule to get from a material consequence to a logical consequence. A,B,C ⊨ D and A,B,C ⊨ D → E allow us to derive A,B,C ⊨ E. * Modus Tollens A different rule to get from a material consequence to a logical consequence. A,B,C ⊨ ¬E and A,B,C ⊨ D → E allow us to derive A,B,C ⊨ ¬D. * Logical Equivalence, Biconditional: A ↔︎ B, A if and only if B, A iff B, A ≡ B, A = B A and B are both true or both false. * Logical conjunction: A and B, A ∧ B, A & B, A && B, A × B Both A and B have to be true for this sentence to be true. * Logical disjunction: A or B, A ∨ B, A | B, A || B, A + B Either A or B or both have to be true for this sentence to be true. * Exclusive or: A xor B, A ⊻ B, A ^ B, A ≢ B, A ⊕ B Either A or B but not both have to be true for this sentence to be true. * Negated logical conjunction, A nand B, A ⊼ B Both A and B have to be false for this sentence to be true. It’s only listed here because every material connective up to here may be written in terms of the nand symbol which is easy to implement in transistor circuitry. Thus computer circuitry logic can closely mirror formal logic. ( true ) = (( A nand A ) nand A) ( false ) = ( (A nand (A nand A ) ) nand (A nand (A nand A ) ) ) ( A → B ) = ( A nand (B nand B) ) ( A = B ) = ( ( A nand B ) nand ( (A nand A ) nand ( B nand B ) ) ) ( A and B ) = ( (A nand B) nand (A nand B) ) ( A or B ) = ( (A nand A) nand (B nand B) ) ( A xor B ) = ( ( A nand B ) nand A ) nand ( ( B nand A ) nand B ) ( not A ) = ( A nand A ) Here are some things to verify: ( A → ( B → A ) ) = ( A nand (( B nand (A nand A) ) nand ( B nand (A nand A) )) ) = (true) ( ( A → ( B → C ) ) → ( ( A → B ) → ( A → C ) ) ) = ( ( A nand ( ( B nand (C nand C) ) nand ( B nand (C nand C) ) ) ) nand ( ( ( A nand (B nand B) ) nand ( ( A nand (C nand C) ) nand ( A nand (C nand C) ) ) ) nand ( ( A nand (B nand B) ) nand ( ( A nand (C nand C) ) nand ( A nand (C nand C) ) ) ) ) ) = ( true ) ( ( ( not A ) → ( not B ) ) → ( B → A ) ) = ( ( (A nand A) nand ( (B nand B) nand (B nand B) ) ) nand ( ( B nand (A nand A) ) nand ( B nand (A nand A) ) ) ) = ( true ) (( A nand ( B nand C ) ) nand ( ( D nand ( D nand D ) ) nand ( ( E nand B ) nand ( ( A nand E ) nand ( A nand E ) ) ) ) ) = ( true ) Sentences which are always true because of their structure are called tautologies. Since tautologies are true regardless of what one assumes, they need nothing on the left side of the logical implications sign. These tautologies are used again and again as axioms ( or theorems depending on what logical axioms one chooses to start with ). Many of them will have formal names. ⊨ ( A → ( B → A ) ) ⊨ ( ( A → ( B → C ) ) → ( ( A → B ) → ( A → C ) ) ) ⊨ ( ( A ∧ B ) → A ) ⊨ ( ( A ∧ B ) → B ) ⊨ ( A → ( B → ( A ∧ B ) ) ) ⊨ ( A → ( A ∨ B ) ) ⊨ ( B → ( A ∨ B ) ) ⊨ ( ( A → C ) → ( ( B → C ) → ( ( A ∨ B ) → C ) ) ) ⊨ ( ( A → B ) → ( ( A → ( ¬ B ) ) → ( ¬ A ) ) ) ⊨ ( A → ( ( ¬ A ) → B ) ) ⊨ ( A ∨ ( ¬ A ) ) ⊨ ( ( A ↔︎ B ) → ( A → B ) ) ⊨ ( ( A ↔︎ B ) → ( B → A ) ) ⊨ ( ( A → B ) → ( ( B → A ) → ( A ↔︎ B ) ) ) Predicate logic is the logic when the sentences are about objects. It introduces new symbols. Here we will use lowercase letters for the objects and upper case letters for the sentences. The Universal quantifier: for all x, ∀x This prefix to a sentence indicates that it is true for all x in the universe of discourse. The Existential quantifier: there exists x, ∃x This prefix indicates that there is at least one x in the universe of discourse that makes the following sentence true. But this is the same thing as saying it is not true that for every x in the universe of discourse that a sentence is false. So ∃x A is a mind-soothing abbreviation for ¬∀x (¬A). Then we can assert our universe contains y by saying ∃x (x=y) instead of ¬∀x (¬(x=y)). Back to Math In addition to logic, two mathematical castles are considered to be large enough to hold most of mathematics: Set Theory and Category Theory. (“Theory” in mathematics, refers to these structures build on axioms and definitions.) Set Theory In set theory, we have a universe of objects called sets which generalize the notion of a bag containing things. Unlike a real bag, the only thing a set can contain is other sets and the only distinguishing property about a set is its contents and the number of those contents may exceed every possible natural number. Thus set theory introduces concepts of membership and infinity. Typically in a computer we only have finite resources so infinite sets aren’t realized in computer hardware. But some of the math computers do relates to representations of objects represented formally by an infinite set. Like the decimal (or binary) expansion of 1/3 never ends but we judge a computer’s math skills based on what answer it gives to 3 × ( 1 / 3 ). Category Theory If set theory is about generalizations about bags, Category theory is about generalizations about objects and arrows and ways to combine arrows. What are the objects and what are the arrows? Almost anything. So a diagram with arrows just might relate to Category Theory. It’s doubtful you will encounter this topic in your first three years. Number Theory The natural numbers run from 0 on up. The product or sum of any two natural numbers is a natural number. Every natural number larger than 0 is expressible as a product of primes — numbers which have no factor other than 1 and themselves. These statement about the positive integers (sometimes the non-negative integers) are the topic of number theory. Do not waste time on the Collatz problem, because that is actually a question about whether algorithms stop which hasn’t been solved because such questions can appear very simple and be very hard to solve. Fermat’s last theorem for example was only recently proven with complex relationships that seem quite far removed from number theory. Analysis Set theory allows us to build infinite sets that represent the real numbers like 0, 1, -1, 1/2, 1/3, √2 and π. Analysis is the study of these real numbers and how we know e^(iπ)+1=0 and (6 + 6/4 + 6/9 + 6/16 + 6/25 + … ) = π². Throughout your Freshman year you will learn differential and integral calculus, which are based in the field of analysis.
There's a difference between giving advice and lecturing. It seems you have steamrolled over the former into the latter. Please Register or Log in to view the hidden image! There also doesn't seem to be much space in your advice for having fun? If I had thrown away all my non-academic books I would had little to help me distress, little to help me take my mind off the academic, to help me relax. Being a freshman is a time to enjoy yourself, to make mistakes, to live and learn.
