Exercises about Sets
... a) Write all of the subset relations that exist between A, B, C, and D. b) Compute A B. Draw a Venn diagram to illustrate. c) Compute A B Draw a Venn diagram to illustrate. d) Compute B C D Draw a Venn diagram to illustrate. e) Compute B C D Draw a Venn diagram to illustrate. f) Compute B ...
... a) Write all of the subset relations that exist between A, B, C, and D. b) Compute A B. Draw a Venn diagram to illustrate. c) Compute A B Draw a Venn diagram to illustrate. d) Compute B C D Draw a Venn diagram to illustrate. e) Compute B C D Draw a Venn diagram to illustrate. f) Compute B ...
Lecture 4: Combinations, Subsets and Multisets
... First think of a set S0 having n different elements: Instead of ai that appears ni times in S, the new set S0 has elements (ai )1 , (ai )2 , . . . , (ai )ni . Then we know that there are P(n, n) = n! permutations of S0 . There are ni ! different permutations of (ai )1 , (ai )2 , . . . , (ai )ni that ...
... First think of a set S0 having n different elements: Instead of ai that appears ni times in S, the new set S0 has elements (ai )1 , (ai )2 , . . . , (ai )ni . Then we know that there are P(n, n) = n! permutations of S0 . There are ni ! different permutations of (ai )1 , (ai )2 , . . . , (ai )ni that ...
Here`s a handout
... • If = 10, then “close to π” means within ten of π (between −6.86 and 13.14). All terms of the sequence are distance less than 10 from π. This same sequence does not converge to, say, 10.2 Let’s see why limn→∞ an 6= 10. ...
... • If = 10, then “close to π” means within ten of π (between −6.86 and 13.14). All terms of the sequence are distance less than 10 from π. This same sequence does not converge to, say, 10.2 Let’s see why limn→∞ an 6= 10. ...
Relations and Functions
... • x-axis: This is the horizontal axis. • y-axis: This is the vertical axis • Origin: This is the center point (0,0) • Each point on the coordinate plane can be represented by an ordered pair (x,y), where x is the distance from Origin on the X-Axis and y is the distance from Origin on the Y-Axis. ...
... • x-axis: This is the horizontal axis. • y-axis: This is the vertical axis • Origin: This is the center point (0,0) • Each point on the coordinate plane can be represented by an ordered pair (x,y), where x is the distance from Origin on the X-Axis and y is the distance from Origin on the Y-Axis. ...
Outline for Chapter 10
... infinite. Sets that are neither finite nor countably infinite are called “uncountable.” Nobody even suspected the existence of uncountable sets until Georg Cantor proved the following theorem, which can be used to show that many sets are uncountable. In particular we can use it to show that the set ...
... infinite. Sets that are neither finite nor countably infinite are called “uncountable.” Nobody even suspected the existence of uncountable sets until Georg Cantor proved the following theorem, which can be used to show that many sets are uncountable. In particular we can use it to show that the set ...
modulo one uniform distribution of the sequence of logarithms of
... and the sufficiency of Weyl's criterion proves the sequence {y.}°° to be uniformly distributed mod 1. Lemma 2. If a is a positive algebraic number not equal to one, then In a is irrational. Proof. Assume, to the contrary, In a = (p/q), where p and q are non-zero integers. Then e p / q = a9 so that e ...
... and the sufficiency of Weyl's criterion proves the sequence {y.}°° to be uniformly distributed mod 1. Lemma 2. If a is a positive algebraic number not equal to one, then In a is irrational. Proof. Assume, to the contrary, In a = (p/q), where p and q are non-zero integers. Then e p / q = a9 so that e ...
Functional Notation
... In the mathematical definition of a function, there is the word unique. This just means that to be function you can’t have an arrow diagram that looks like the following: ...
... In the mathematical definition of a function, there is the word unique. This just means that to be function you can’t have an arrow diagram that looks like the following: ...
Slide 1
... The initial conditions: the values of the first few terms a0, a1, … Example: For all integers k ≥ 2, find the terms b2, b3 and b4: bk = bk-1 + bk-2 (recurrence relation) ...
... The initial conditions: the values of the first few terms a0, a1, … Example: For all integers k ≥ 2, find the terms b2, b3 and b4: bk = bk-1 + bk-2 (recurrence relation) ...
9.2 Summation Notation
... It is an arithmetic sequence with first term a = 1 and common difference d = 1. ...
... It is an arithmetic sequence with first term a = 1 and common difference d = 1. ...
Full text
... Case 1: n = 2\ with / > 1. It Is clear that M = 2 = v(ajaj), where j = 2'"1. Thus, by (3.1), v(a„) = l = S(n). Case 2% n = 2ei +2*2 +>-+2et, with 0
... Case 1: n = 2\ with / > 1. It Is clear that M = 2 = v(ajaj), where j = 2'"1. Thus, by (3.1), v(a„) = l = S(n). Case 2% n = 2ei +2*2 +>-+2et, with 0
