Sets, Infinity, and Mappings - University of Southern California
... b) Use part (a) to prove that z is irrational. Hint: A number is rational if and only if its decimal expansion has an eventually repeating pattern. Suppose {a11 , a22 , a33 , . . .} has an eventually repeating pattern. G. Infinite levels of infinity A finite set A has strictly smaller cardinality th ...
... b) Use part (a) to prove that z is irrational. Hint: A number is rational if and only if its decimal expansion has an eventually repeating pattern. Suppose {a11 , a22 , a33 , . . .} has an eventually repeating pattern. G. Infinite levels of infinity A finite set A has strictly smaller cardinality th ...
PIGEONHOLE PRINCIPLE
... To prove the minimal property of 33, we have to construct a graph with 32 edges such that it has no red triangle or blue triangle. From the first part of the proof, we see that there should not be a complete subgraph of 6 vertices if no red or blue triangle appears. To avoid such a complete subgraph ...
... To prove the minimal property of 33, we have to construct a graph with 32 edges such that it has no red triangle or blue triangle. From the first part of the proof, we see that there should not be a complete subgraph of 6 vertices if no red or blue triangle appears. To avoid such a complete subgraph ...
Daftar simbol matematika
... or join in a lattice or propositional logic, lattice theory exclusive or ...
... or join in a lattice or propositional logic, lattice theory exclusive or ...
MA310 MAPLE — Example Sheet 2 1 Solving Equations
... Being a computer system for doing mathematics, Maple supports a variety of data structures, such as lists, arrays or tables as well as mathematical objects like sets, polynomials, vectors or matrices. Of particular importance are sequences and lists. A sequence is just a collection of expressions, s ...
... Being a computer system for doing mathematics, Maple supports a variety of data structures, such as lists, arrays or tables as well as mathematical objects like sets, polynomials, vectors or matrices. Of particular importance are sequences and lists. A sequence is just a collection of expressions, s ...
![Maximum subsets of (0,1] with no solutions to x](http://s1.studyres.com/store/data/004884619_1-aef9649f88c5cc5f6ca0274124904717-300x300.png)
Integral calculus, and introduction to analysis
... receptionist wants to know how many rooms are occupied, s/he doesn’t have to visit all the rooms to check s/he can just count the number of hooks whose keys are missing. There is nothing deep about this example, but it illustrates a point that is important in mathematics. In the example, the occupie ...
... receptionist wants to know how many rooms are occupied, s/he doesn’t have to visit all the rooms to check s/he can just count the number of hooks whose keys are missing. There is nothing deep about this example, but it illustrates a point that is important in mathematics. In the example, the occupie ...
Lecture Slides
... Find the i-th smallest element on the low side if i < k Find the (i-k)-th smallest element on the high side if i > k ...
... Find the i-th smallest element on the low side if i < k Find the (i-k)-th smallest element on the high side if i > k ...
BB Chapter 2 - WordPress.com
... Manipulatives Suppose you have 78 number tiles. Describe how to illustrate 78 ÷13 with the tiles, using each of the three basic conceptual models for division. a. Repeated subtraction. Remove groups of 13 tiles each. Since 6 groups are formed 78 ÷ 13 = 6. b. Partition. Partition the tiles into 13 eq ...
... Manipulatives Suppose you have 78 number tiles. Describe how to illustrate 78 ÷13 with the tiles, using each of the three basic conceptual models for division. a. Repeated subtraction. Remove groups of 13 tiles each. Since 6 groups are formed 78 ÷ 13 = 6. b. Partition. Partition the tiles into 13 eq ...
psychology - NIILM University
... (v) If each variable and constant of the conditional can be assigned a T or F, then the conditional is not a tautology, for the assignment constitutes a row of the complete truth table under which the conditional is false, i.e., the conditional is not a tautology, i.e., the corresponding argument is ...
... (v) If each variable and constant of the conditional can be assigned a T or F, then the conditional is not a tautology, for the assignment constitutes a row of the complete truth table under which the conditional is false, i.e., the conditional is not a tautology, i.e., the corresponding argument is ...
Lesson 2-1 part 1 Powerpoint - peacock
... Every point on a vertical line has the same xcoordinate, so a vertical line cannot represent a function. If a vertical line passes through more than one point on the graph of a relation, the relation must have more than one point with the same x-coordinate. Therefore the relation is not a function. ...
... Every point on a vertical line has the same xcoordinate, so a vertical line cannot represent a function. If a vertical line passes through more than one point on the graph of a relation, the relation must have more than one point with the same x-coordinate. Therefore the relation is not a function. ...
Document
... Set C: Number Distinct Proper Subsets = 26 – 1 = 63 Set D: Number Distinct Proper Subsets = 21 – 1 = 1 Set E: Number Distinct Proper Subsets = 20 – 1 = 0 Based on this you can see that the Number of Distinct Proper Subsets for any set is 2n – 1 where n is number elements. ...
... Set C: Number Distinct Proper Subsets = 26 – 1 = 63 Set D: Number Distinct Proper Subsets = 21 – 1 = 1 Set E: Number Distinct Proper Subsets = 20 – 1 = 0 Based on this you can see that the Number of Distinct Proper Subsets for any set is 2n – 1 where n is number elements. ...
introduction to proofs
... For the other part first consider a divisor d of 1 that is a positive integer. By the prior item 1 ≥ d so the only such divisor is d = 1. Clearly 0 is not a divisor of 1 so the only remaining candidates are negative integers. We have shown that for d , a ∈ Z, if d | a then −d | a, so if a negative n ...
... For the other part first consider a divisor d of 1 that is a positive integer. By the prior item 1 ≥ d so the only such divisor is d = 1. Clearly 0 is not a divisor of 1 so the only remaining candidates are negative integers. We have shown that for d , a ∈ Z, if d | a then −d | a, so if a negative n ...
Countable or Uncountable*That is the question!
... If A is a countably infinite set and B is a subset of A then B is countable. Case I: If B is the empty set or a finite set then B is countable. Case II: B is an infinite set Since A is countable we can write the elements of A in the order a1, a2, a3, . . . If B is a subset of A then an infinite num ...
... If A is a countably infinite set and B is a subset of A then B is countable. Case I: If B is the empty set or a finite set then B is countable. Case II: B is an infinite set Since A is countable we can write the elements of A in the order a1, a2, a3, . . . If B is a subset of A then an infinite num ...
algebraic topology - School of Mathematics, TIFR
... Example 2.4 Let G = Z/(m) where m is any integer ≥ 0. Set k̄ + ¯l = k + l. It is easy to check that this defines an operation which satisfies our axioms. G becomes thus an abelian group and is finite if m > 0. Example 2.5 The non-zero real numbers denoted by R∗ (resp. the non-zero rational numbers d ...
... Example 2.4 Let G = Z/(m) where m is any integer ≥ 0. Set k̄ + ¯l = k + l. It is easy to check that this defines an operation which satisfies our axioms. G becomes thus an abelian group and is finite if m > 0. Example 2.5 The non-zero real numbers denoted by R∗ (resp. the non-zero rational numbers d ...
MATH 337 Cardinality
... The Schroeder-Bernstein Theorem states that if A1 ⊆ A2 ⊆ A3 and A1 is equivalent to A3 , then A2 is equivalent to A3 . We can apply the result to show that the open interval We simply note that (0, 1) is equivalent to the half-open interval [0,1) . [1 / 2,1) ⊆ (0, 1) ⊆ [0, 1) and [1 / 2,1) ~ [0,1) b ...
... The Schroeder-Bernstein Theorem states that if A1 ⊆ A2 ⊆ A3 and A1 is equivalent to A3 , then A2 is equivalent to A3 . We can apply the result to show that the open interval We simply note that (0, 1) is equivalent to the half-open interval [0,1) . [1 / 2,1) ⊆ (0, 1) ⊆ [0, 1) and [1 / 2,1) ~ [0,1) b ...
Countable and Uncountable Sets What follows is a different, and I
... Any two finite sets with the same number of elements can be put into 1-1 correspondence. Conversely, if A and B are finite sets that can be put into 1-1 correspondence, then we know A and B have the same number of elements. In what follows, the idea of the “size” of a set is extended to infinite sets. ...
... Any two finite sets with the same number of elements can be put into 1-1 correspondence. Conversely, if A and B are finite sets that can be put into 1-1 correspondence, then we know A and B have the same number of elements. In what follows, the idea of the “size” of a set is extended to infinite sets. ...