MATH 1113 Review Sheet for the Final Exam
... Inverse Function of a function; connection between domains and ranges of these functions Defining the terms one to one and one to one function o Intuitively, one to one means no partner sharing o Determining when a graph that represents a function is one to one; the horizontal line test Relati ...
... Inverse Function of a function; connection between domains and ranges of these functions Defining the terms one to one and one to one function o Intuitively, one to one means no partner sharing o Determining when a graph that represents a function is one to one; the horizontal line test Relati ...
Math 319 Solutions to Homework 8
... (b) Suppose that xn ≥ 0 for all n and you are told that ((−1)n xn ) converges. Show that (xn ) converges. What is its limit? Let (yn ) := ((−1)n xn ) have limit y. Then its subsequence (y2k ) also has limit y (by 3.4.2). Since y2k = xk ≥ 0 for all k, we have y ≥ 0 by 3.2.4. Similarly y2k+1 converges ...
... (b) Suppose that xn ≥ 0 for all n and you are told that ((−1)n xn ) converges. Show that (xn ) converges. What is its limit? Let (yn ) := ((−1)n xn ) have limit y. Then its subsequence (y2k ) also has limit y (by 3.4.2). Since y2k = xk ≥ 0 for all k, we have y ≥ 0 by 3.2.4. Similarly y2k+1 converges ...
Prove
... To prove a theorem (proposition, lemma) of the form p ⇒ q, we often attempt to build a bridge of implications p ⇒ p1 ⇒ p2 ⇒ · · · ⇒ q2 ⇒ q1 ⇒ q. To obtain such a bridge, we might try to use deductive reasoning (”what can we conclude from p?”) to get p1 from p, then again to get p2 from p1 , and so o ...
... To prove a theorem (proposition, lemma) of the form p ⇒ q, we often attempt to build a bridge of implications p ⇒ p1 ⇒ p2 ⇒ · · · ⇒ q2 ⇒ q1 ⇒ q. To obtain such a bridge, we might try to use deductive reasoning (”what can we conclude from p?”) to get p1 from p, then again to get p2 from p1 , and so o ...
DAVID ESSNER EXAM IV 1984-85
... Bill. If Bill took 6 minutes and Joe took 5 minutes for the race, what was the rate of Bill (to the nearest integer) in feet per minute? (a) 3 (b) 5 (c) 7 (d) 8 (e) 9 16. Let an be a sequence such that a1 = 1, a4 = 13 and an = an-1 + an-2 for n 3. Then a6 = (a) 18 (b) 26 (c) 33 (d) many possible a ...
... Bill. If Bill took 6 minutes and Joe took 5 minutes for the race, what was the rate of Bill (to the nearest integer) in feet per minute? (a) 3 (b) 5 (c) 7 (d) 8 (e) 9 16. Let an be a sequence such that a1 = 1, a4 = 13 and an = an-1 + an-2 for n 3. Then a6 = (a) 18 (b) 26 (c) 33 (d) many possible a ...
1 Sets, Set Construction, and Subsets
... on the left of the colon, we now have a set that elements are coming from on the right. Furthermore, we now have a formula on the left rather than a property on the right. The difference is that for a property, when we plug in an element from the given set, we either obtain a true or false value, bu ...
... on the left of the colon, we now have a set that elements are coming from on the right. Furthermore, we now have a formula on the left rather than a property on the right. The difference is that for a property, when we plug in an element from the given set, we either obtain a true or false value, bu ...
Problems 98 - Abelkonkurransen
... 19. Consider a ping-pong match between two teams, each consisting of 1000 players. Each player played against each player of the other team exactly once (there are no draws in ping-pong). Prove that there exist ten players, all from the same team, such that every member of the other team has lost hi ...
... 19. Consider a ping-pong match between two teams, each consisting of 1000 players. Each player played against each player of the other team exactly once (there are no draws in ping-pong). Prove that there exist ten players, all from the same team, such that every member of the other team has lost hi ...
