Math 3000 Section 003 Intro to Abstract Math Midterm 1

Transcription

1 Math 3000 Section 003 Intro to Abstract Math Midterm 1 Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Name: Points: Read all problems carefully and write your solutions essay-style using full sentences. In addition, please remember to always state whether you are giving a proof or a counterexample, and what type of proof (direct proof, proof by contrapositive, contradiction, or cases) you are using. 1. The following table gives several possibilities to begin a proof of the statement that P Q. For each beginning, check whether the opening sentence is a correct beginning and identify the type of proof that is being used; otherwise indicate that a mistake must have been made. First step to prove that P Q direct contrapositive contradiction mistake Assume that P is true. Assume that P is false. Assume that Q is true. Assume that Q is false. Assume that P is true and Q is true. Assume that P is true and Q is false. Assume that P is false and Q is true. Assume that P is false and Q is false. Assume that P Q is true. Assume that P Q is false. 2. Let a and b be integers and define the following concepts. Use correct mathematical notation. (a) a divides b (b) a and b have the same parity 3. Let S be a set and define the following concepts. Use correct mathematical notation if needed. (a) a partition of S (b) the power set P(S)

2 Math 3000 Section 003 Intro to Abstract Math Midterm 1, UC Denver, Spring Prove or disprove: If a and b are two even nonnegative integers, then a b is even. 5. Prove or disprove: There exists a largest positive integer that is divisible by 2012.

3 Math 3000 Section 003 Intro to Abstract Math Midterm 1, UC Denver, Spring For every positive real number, there exists a positive integer whose reciprocal is smaller. (a) Express the above statement in symbols. (b) Decide whether the statement is true or false, and supply a proof or counterexample.

4 Math 3000 Section 003 Intro to Abstract Math Midterm 1, UC Denver, Spring Let x = (x 1, x 2 ) R 2 and y = (y 1, y 2 ) R 2 be two elements in the Cartesian product R R (two-dimensional vectors). We say that x dominates y if (and only if) x 1 > y 1 and x 2 > y 2. (a) Which of the following vectors is dominated by x = (3000, 3200)? y = (3000, 3195) y = (2421, 3250) y = (1401, 2411) y = (1010, 8990) (b) Prove or disprove: If x 1 + x 2 > y 1 + y 2, then x dominates y. (c) Formulate the converse of the statement in (b), and prove it. (d) Formulate the inverse of the statement in (b), and prove it.

5 Math 3000 Section 003 Intro to Abstract Math Midterm 1, UC Denver, Spring Let X and Y be two finite sets. (a) Prove the following triangle-type inequality: X Y X + Y. (b) Conjecture a similar relationship between X Y and X Y and prove your assertion. [Hint: Consider the set (X Y ) Y and use the result that you have proven in part (a).]

6 Math 3000 Section 003 Intro to Abstract Math Midterm 1, UC Denver, Spring Prove or disprove: If S and T are two sets, then P(S) P(T ) P(S T ).

7 Math 3000 Section 003 Intro to Abstract Math Midterm 1, UC Denver, Spring Read the following poem If by Rudyard Kipling ( ) and then check all (but only those) of the below statements that are logically equivalent to the related verses in this poem. If you can keep your head when all about you Are losing theirs and blaming it on you; If you can trust yourself when all men doubt you, But make allowance for their doubting too; If you can wait and not be tired by waiting, Or being lied about, don t deal in lies, Or being hated, don t give way to hating, And yet don t look too good, nor talk too wise: If you can dream and not make dreams your master; If you can think and not make thoughts your aim; If you can meet with Triumph and Disaster And treat those two imposters just the same; If you can bear to hear the truth you ve spoken Twisted by knaves to make a trap for fools, Or watch the things you gave your life to, broken, And stoop and build em up with worn-out tools; If you can make one heap of all your winnings And risk it on one turn of pitch-and-toss, And lose, and start again at your beginnings And never breathe a word about your loss; If you can force your heart and nerve and sinew To serve your turn long after they are gone, And so hold on when there is nothing in you Except the Will which says to them: Hold on! If you can talk with crowds and keep your virtue, Or walk with kings nor lose the common touch, If neither foes nor loving friends can hurt you, If all men count with you, but none too much; If you can fill the unforgiving minute With sixty seconds worth of distance run Yours is the Earth and everything that s in it, And which is more you ll be a Man, my son! Yours is the Earth and everything that s in it only if you can keep your head when all about you are losing theirs and blaming it on you. You ll be a Man if you look good and talk wise. The ability to dream and think is necessary to be a Man. Either you can dream and think, or the Earth is not yours. You won t be a Man if you cannot meet with Triumph and Disaster. If yours is the Earth and everything that s in it, then you can make one heap of all your winnings and risk it on one turn of pitch-and-toss, and lose, and start again at your beginnings and never breathe a word about your loss. Either you ll be a Man, or either foes or loving friends can hurt you. Yours being the Earth suffices for your ability to fill the unforgiving minute with sixty seconds worth of distance run.

8 Math 3000 Section 003 Intro to Abstract Math Midterm 1, UC Denver, Spring This is a bonus problem that you should work only if you have completed all other problems. Any pair of two successive odd primes (2k 1, 2k + 1) for some positive integer k is called a twin prime: examples are (3,5), (5,7), (11,13), (17,19), (29,31), (41,43), etc. The twin prime conjecture states that there are infinitely many twin primes - and this is still an open question! Here is an easier problem: Prove that there exists a unique triple of three successive odd primes. [Hint: Find the triple, and then use congruences with respect to a suitable modulus].