Lecture notes 2 -- Sets
... Now, instead of repeating all of these numbers each time we want to refer to this set, we can just reference S. Traditionally we use upper-case letters as names for sets, but this rule is not set in stone. The set S has five elements: the numbers 1, 2, 3, 4, and 5. The number 7, 100, and 3.14 are no ...
... Now, instead of repeating all of these numbers each time we want to refer to this set, we can just reference S. Traditionally we use upper-case letters as names for sets, but this rule is not set in stone. The set S has five elements: the numbers 1, 2, 3, 4, and 5. The number 7, 100, and 3.14 are no ...
Sets and Functions
... How can I decide from a graph whether it is a function? The vertical line test is a way to determine whether or not we have a function. If a vertical line intersects the graph in more than one place, then it is NOT a function. The test is simply a restatement of the definition of a function which st ...
... How can I decide from a graph whether it is a function? The vertical line test is a way to determine whether or not we have a function. If a vertical line intersects the graph in more than one place, then it is NOT a function. The test is simply a restatement of the definition of a function which st ...
Set Notation File
... inequalities, because for some reason saying "x < 3" isn't good enough, so instead they'll want you to phrase the answer as "the solution set is { x | x is a real number and x < 3 }". How this adds anything to the student's understanding, I don't know. But I digress.... A set, informally, is a colle ...
... inequalities, because for some reason saying "x < 3" isn't good enough, so instead they'll want you to phrase the answer as "the solution set is { x | x is a real number and x < 3 }". How this adds anything to the student's understanding, I don't know. But I digress.... A set, informally, is a colle ...
Proposition: The following properties hold A ∩ B ⊆ A, A ∩ B ⊆ B, A
... X − A = {x | x ∈ X and x 6∈ A}. Note: In applications there is usually a universal set in the background denoted U . This could be the set of all the people in the world, the set of all real numbers, etc., Definition: If there is a universal set U , we define the complement of A to be ∼A = U − A = { ...
... X − A = {x | x ∈ X and x 6∈ A}. Note: In applications there is usually a universal set in the background denoted U . This could be the set of all the people in the world, the set of all real numbers, etc., Definition: If there is a universal set U , we define the complement of A to be ∼A = U − A = { ...
BASIC SET THEORY
... the singleton set containing x . THEOREM: {x } = {y } if and only if x = y . Proof: If {x } = {y }, then: Any member of {y } is also a member of {x }. Any member of {x } is equal to x itself. So any member of {y } is equal to x itself. So {y } is the singleton set {x }. The first several weeks of a ...
... the singleton set containing x . THEOREM: {x } = {y } if and only if x = y . Proof: If {x } = {y }, then: Any member of {y } is also a member of {x }. Any member of {x } is equal to x itself. So any member of {y } is equal to x itself. So {y } is the singleton set {x }. The first several weeks of a ...
Lec2Logic
... -For all x and for all y if x is positive and y is negative then their product must be negative. -The product of a positive and a negative real number is negative. Translate this sentence into a logical expressions. “If a person is female and is a parent, then she is someone’s mother.” F(x) is “x is ...
... -For all x and for all y if x is positive and y is negative then their product must be negative. -The product of a positive and a negative real number is negative. Translate this sentence into a logical expressions. “If a person is female and is a parent, then she is someone’s mother.” F(x) is “x is ...
Sets and Functions
... Also called mapping or transformation … (As an example, if the function f is age, then it “maps” each student from set A To an integer from B to like age (Bob) = 19, age (Alice) = 21 …} ...
... Also called mapping or transformation … (As an example, if the function f is age, then it “maps” each student from set A To an integer from B to like age (Bob) = 19, age (Alice) = 21 …} ...
Chapter 1
... subset of B, symbolized as A B, if and only if each element of A is also an element of B. 2.1.1.4.2. Venn diagram: representation of sets using circles, where the elements of a set are contained within a circle 2.1.1.4.3. Definition of a proper subset of a set: For all sets A and B, A is a proper ...
... subset of B, symbolized as A B, if and only if each element of A is also an element of B. 2.1.1.4.2. Venn diagram: representation of sets using circles, where the elements of a set are contained within a circle 2.1.1.4.3. Definition of a proper subset of a set: For all sets A and B, A is a proper ...
3-8 Unions and Intersection of Sets
... In your left pocket, you have a quarter, a paper clip, and a key. In your right pocket, you have a penny, a quarter, a pencil, and a marble. What is a set that represents the different items in your pockets ...
... In your left pocket, you have a quarter, a paper clip, and a key. In your right pocket, you have a penny, a quarter, a pencil, and a marble. What is a set that represents the different items in your pockets ...
Sets, Functions, Relations - Department of Mathematics
... Exercise: Prove by induction that if |A| = n then |P(A)| = 2n . Multisets. Two ordinary sets are identical if they have the same elements, so for instance, {a, a, b} and {a, b} are the same set because they have exactly the same elements, namely a and b. However, in some applications it might be use ...
... Exercise: Prove by induction that if |A| = n then |P(A)| = 2n . Multisets. Two ordinary sets are identical if they have the same elements, so for instance, {a, a, b} and {a, b} are the same set because they have exactly the same elements, namely a and b. However, in some applications it might be use ...
Set Concepts
... Sets are collections of "things" that are called elements. We looked at sets in our discussion of the Real Numbers, such as the Counting Numbers f1; 2; 3; 4; 5; :::g. Sets can be a collection of any kinds of "things" that we will call ...
... Sets are collections of "things" that are called elements. We looked at sets in our discussion of the Real Numbers, such as the Counting Numbers f1; 2; 3; 4; 5; :::g. Sets can be a collection of any kinds of "things" that we will call ...
A set is a collection of objects. The objects are called elements of the
... A set may have infinitely many elements, so we can’t list all of them. For example let E = {all even integers greater than or equal to 1}. We write this as E = {2, 4, 6, . . . }, where “. . . ” should be read as “et cetera”. When we place an element after the dots, as in K = {2, 4, 6, . . . , 100}, ...
... A set may have infinitely many elements, so we can’t list all of them. For example let E = {all even integers greater than or equal to 1}. We write this as E = {2, 4, 6, . . . }, where “. . . ” should be read as “et cetera”. When we place an element after the dots, as in K = {2, 4, 6, . . . , 100}, ...