Sets

Question 1

Given A and B are subsets of a universet set U.

What does A ∪ B mean? What are the elements in it?

Answer 1

It means A or B. It includes elements that are in at least one of A or B

Question 2

Given A and B are subsets of a universet set U.

What does A ∩ B mean? What are the elements in it?

Answer 2

It means A and B. It includes elements that are in at both A and B

Question 3

Given A and B are subsets of a universet set U.

What does B - A mean? What are the elements in it?

Answer 3

It means the difference of B minus A. It includes elements that are only in B and not A

Question 4

Given A and B are subsets of a universet set U.

What does A^c mean? What are the elements in it

Answer 4

It means the complement of A. It includes elements of U that are not in A.

Note this is different from B-A as B-A is elements of B that are not in A. A^c means elements in U and not in A

Question 5

Answer 5

Question 6

Answer 6

Question 7

How is the following read?

Answer 7

Supposing U is the universal subset. It is read as “x is an element of U such that x is also an element of A_i for at least one i=0,1,2,…,n”.
Basically the union of the set A₀ to A_n

Question 8

How is the following read?

Answer 8

Supposing U is the universal subset. It is read as “x is an element of U such that x is also an element of A_i for atleast one nonnegative integer i}.
Basically the union of the set A₀ to A_∞

Question 9

How is the following read?

Answer 9

Supposing U is the universal subset. It is read as “x is an element of U such that x is also an element of A_i for every i=0,1,2,…,n”.
Basically the intersection of the set A₀ to A_n

Question 10

How is the following read?

Answer 10

Supposing U is the universal subset. It is read as “x is an element of U such that x is also an element of A_i for every one nonnegative integer i}.
Basically the intersection of the set A₀ to A_∞

Question 11

Answer 11

Question 12

Answer 12

Question 13

What does ∅ mean?

Answer 13

Empty set (null set). A set with no elements

Question 14

Answer 14

Question 15

When are two sets called disjoint?

Answer 15

Question 16

Answer 16

Question 17

Given an arbitrary number of sets A₁, A₂, A₃,… When are these sets mutally disjoint?

Answer 17

Basically when none of those sets share any elements

Question 18

Answer 18

Question 19

What is partition?

Answer 19

When sets are divided into nonoverlapping (or disjoint) pieces

Question 20

Given the following image, what can be said about A₁, A₂, A₃, A₄ in relation to A

Answer 20

A₁, A₂, A₃, A₄ are subsets of A meaning A = A₁ ∪ A₂ ∪ A₃ ∪ A₄

Question 21

Just revision

Question 22

Answer 22

Question 23

Answer 23

Basically integers in T₀ are divisible by 3; Integers in T₁ are divisible by 3 with a remainder of 1; Integers in T₂ are divisible by 3 with a remainder of 2. But the integer cannot be all three.

Question 24

What is a power set?

Answer 24

Question 25

What is the unique set that is a subset of every single set

Answer 25

The empty set ∅.

Question 26

Answer 26

Question 27

What is an identity?

Answer 27

An equation that is universally true for all elements in some set. ## Footnote For example, the equation a + b = b + a is an identity for real numbers because it is true for all real numbers a and b

Question 28

If a set, A has n elements, then how many elements does the power set of a have?

Answer 28

2ⁿ elements

Question 29

Answer 29

Question 30

Answer 30

Question 31

What are the two types of set proofs?

Answer 31

1. algebraic proof: using set identities 2. element proof: making use of logic and how sets work. For example A - (A ∩ B) = A - B. You can see this is true because the left side means everything that is not in A and B but in A. The right side is all the elements of A take away all the sets of B. Both sides mean 'only A'