Mathematical Thinking and Set Theory Flashcards
Why do we need mathematical languages?
They provide necessary abstraction to understand a complex phenomenon
What is a variable?
You can think of variables as placeholders when you want to talk about something but either it has one or more values you don’t know or you want whatever you say about it to be equally true for all elements in a given set
What types pf statements do we know and what are their definitions?
- Universal (for all, for each,)
- -> says that a certain property is true for alll elements in a set - Conditional (if p then q)
- -> if one thing is true then another thing also gotta be true - Existential(for some, there exists, there is at least one thing for which the property is true
Let A = {1, 2, 3} and B = {1, 1, 2, 3, 3, 3} What are their elements and how are they related?
They have the same three elements: 1, 2, 3. Therefore, they are just different ways to represent the same sets.
How is a subset defined?
If A and B are sets, then A is called a subset of B if every element of A is also in B
How is a proper subset defined?
If A and B are sets, then A is called a subset of B if every element of A is also in B but there is at least one element in B that is not in A
What is a ordered pair?
Given elements a and b, (a, b) denotes the ordered pair consisting of a and b with a being the first and b the second element. Two ordered pairs (a, b) and (c, d) are equal if a = c and b = d
What is the cartesian product of two sets A and B?
The cartesian product of A and B: A x B “speak cross” is the set of all ordered pairs (a, b) where a is in A and b is in B.
When are two sets equal?
If every element fo A is in B and every element in B is in A.
is {0} equal to 0?
no, because it is a set with one element (0) whereas 0 is just a number that represents the number zero-
What is the Power Set of the set of A?
the power set of A is the set of all subsets of A
i.e. P({x, y})
= {empty set, {x}, {y}, {x, y})
