1.3 Variables and Sets Flashcards
What is a variable, what can it stand for and how can they be combined with other variables?
A variable is an object that is represented by a letter.
A variable can stand for anything, a number, a person, etc.
Statements using variables can be combined using connectives, just like statements without variables.
What is a truth set and why do we need it?
The truth set of a statement P(x) is the set of all values of x that make the statement P(x) true.
Truth set of P(x) = {x | P(x)}.
We need truth sets because statements with variables can be true or false depending on the value of the variable.
What is a set?
A set is a collection of objects.
What is an element of a set and how do we denote something is an element of a set?
The elements of a set are the objects in the collection, denoted with the symbol ∈.
How can we define a set with the numbers 3, 7, 14?
If we represent the above set with the letter, A, how can we say 7 is an element of A?
A = {3, 7, 14}
7 ∈ A
What is this called and what does it mean?
B = {x | x is a prime number}
This is called an elementhood test.
An elementhood test for a set is a condition that an item needs to meet in order to be considered an element in the set.
B = the set of all x such that x is a prime number
In the statement:
y ∈ {x | x^2 < 9}
What is the free variable and what is the bound/dummy variable?
y is the free variable. It is free to stand for anything you want to plug in. The free variables in a statement stand for objects that the statement says something about.
x is the bound/dummy variable. They are simply letters to help express an idea and doesn’t stand for a particular object. The bound variable symbol can always be replaced by another symbol, or eliminated altogether, and the meaning is the same.
How do you bind a variable x?
You bind the variable x with the notation {x|…}
What is the type of expression for {x| P(x)} and y ∈ {x | P(x)}?
{x | P(x)} is the name for a set.
y ∈ {x | P(x)} is a statement.
Note that y ∈ {x | P(x)} is the same as P(y), which is a statement about y but not x.
If A is the truth set of the statement P(x) and y ∈ {x | P(x)}, express the relationship between A and y?
y ∈ A = P(y)
What is a universe of discourse and what are the most important ones?
The universe of discourse is the set of all possible values for a variable in a statement.
The most important ones are:
R = set of all real numbers, any number on the number line
Q = set of all rational numbers, any number that can be written as a fraction p/q, where p and w are integers
Z = set of all integers
N = set of natural numbers, all positive integers. Some books include/don’t include 0 as a natural number
What does this mean: {x ∈︎ U | P(x)}?
The set of all x in U such that P(x).
Only elements of U are to be considered for elementhood in this truth set.
And among those elements of U, only those that pass the elementhood test P(x) will actually be in the truth set.
How do you write y ∈︎ {x ∈︎ A | P(x)} using logical connectives?
y ∈︎ A ^ P(y)
What is the definition of a truth set tautology?
If P(x) is true for every x ∈︎ U, then the truth set of P(x) is the whole universe U.
What is the definition of a truth set contradiction ?
If P(x) is false for every x ∈︎ U, then nothing in U can pass the elementhood test. So this set will be empty, or will be the empty set, ∅︎.
