A group is, underneath everything else this course adds, a set with one operation attached to it. Every idea below is already familiar from Discrete Math I. This section only puts the vocabulary back in your hands before it starts carrying real weight. Three later ideas lean on it directly. A subgroup, starting in chapter 6, is just a subset that survives the operation without leaving it. A direct product, in chapter 11, is built straight from the Cartesian product G \times H. A binary operation itself, starting in chapter 1, is a function whose domain is G \times G — a set you cannot even write down without the Cartesian product reviewed here.
Think of a set as a guest list. A subset is a smaller list drawn entirely from the first one; nobody on it is a stranger to the bigger list. Union merges two guest lists into one, counting each person once even if they appear on both. Intersection keeps only the names shared by both lists. The Cartesian product works differently. Instead of merging two lists into one bigger list, it pairs every name on list A with every name on list B. The result is a list of pairings, not a list of names. A seating chart matching every diner to every table is exactly this kind of pairing. Swap the diner and the table and you get a different pairing, not the same one back.
A set is an unordered collection of distinct objects. A set operation is a rule that builds a new set out of one or two existing ones. Union, intersection, difference, and the Cartesian product below are the four this course actually reaches for. For a set A, B \subset A means every element of B is also an element of A. The union A \cup B collects everything in A or B, or both. The intersection A \cap B keeps only what's in both. The difference A \setminus B keeps what's in A and removes anything also in B. The Cartesian product A \times B is the set of all ordered pairs (a,b) with a \in A and b \in B. Order matters here: A \times B and B \times A contain different pairs unless A = B. Starting in chapter 6, you will constantly ask two separate questions together. Is this candidate actually a subset? And is it closed under some operation? Two standing facts are worth stating outright. The empty set \varnothing is a subset of every set, and every set is a subset of itself. Both follow straight from the definition of subset above. There is no element of \varnothing, or of A itself, that could ever fail the "also in A" test.
Theorem
The Empty Set is a Subset of Every Set
also known as: vacuous subset
Recalled from College Algebra · Sets, plainly — first taught there.
For every set A, \varnothing \subset A.
B \subset A is false only when B contains some element that A lacks. Take B = \varnothing: it contains no elements at all, so there is nothing it could possibly contain that A doesn't also contain. The condition for B \subset A to fail can never be met, so \varnothing \subset A holds automatically — for every set A, including A = \varnothing itself.
Shade the difference
Click every region belonging to A but not to B.
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Subset or not?
Sort each candidate set by whether it is a subset of {1,2,3,4}.
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Worked example
Union, intersection, and difference of two finite sets
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1
Let A = \{1, 2, 3, 4\} and B = \{3, 4, 5\}. Find A \cup B, A \cap B, and A \setminus B.
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2
A \cup B collects every element appearing in either set, listed once each: \{1,2,3,4,5\}.
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3
A \cap B keeps only 3 and 4, since those are the only elements both sets share.
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4\boxed{A \cup B = \{1,2,3,4,5\}, \quad A \cap B = \{3,4\}, \quad A \setminus B = \{1,2\}}5 is in B but not A, so it never appears in A \ B — the order in "difference" matters
5 is in B but not A, so it never appears in A \ B — the order in "difference" matters
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Worked example
A Cartesian product as the domain of an operation
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1
Let A = \{1, 2, 3\}. List A \times A, and say how many ordered pairs it contains.
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2
Every element of the first A pairs with every element of the second: (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3).
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3\boxed{|A \times A| = 9}
This is exactly the set a binary operation on A would need as its domain — chapter 1 defines a binary operation as a function A \times A \to A, one output for every one of these nine input pairs.
Where this goes next
Chapter 6: Subgroups
The subgroup test is a subset question wearing algebra's clothes. Given a group G and a subset H \subset G, chapter 6 asks three things: is H closed under G's operation, does it contain the identity, and does it contain every element's inverse? Each check is really a membership question. Does this particular element of G \times G's output land back inside H? You can decide that with nothing more than the subset and closure ideas reviewed here.
Quick review
Practice from earlier coursework, pulled in for this topic.
From Discrete Mathematics I — Logic, Proof, and Structure
What does {x in Q : x^2 = 2} equal?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Why
sqrt(2) is irrational, so no rational number satisfies x^2=2 -- the rule describes the empty set.
From Discrete Mathematics I — Logic, Proof, and Structure
Why is set-builder notation necessary for a set like the real numbers between 0 and 1?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Why
An infinite set cannot be written out element by element -- a rule is the only way to name it precisely.
Quick review
Practice from earlier coursework, pulled in for this topic.
From Discrete Mathematics I — Logic, Proof, and Structure
Is {1,5} a subset of {1,2,3,4}?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Why
5 is in {1,5} but not in {1,2,3,4} -- one missing element is enough to fail the subset claim.
From Discrete Mathematics I — Logic, Proof, and Structure
What does A subseteq B mean?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Why
A subseteq B means every element of A is guaranteed to also be an element of B.
Quick review
Practice from earlier coursework, pulled in for this topic.
From Discrete Mathematics I — Logic, Proof, and Structure
A student claims A setminus B always equals B setminus A. Test with A={1,2}, B={2,3} -- is the claim correct?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Why
The two differences pick out different leftover elements -- set difference is not commutative in general.
From Discrete Mathematics I — Logic, Proof, and Structure
Two sets are called disjoint exactly when...
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Why
Disjoint sets share no elements at all -- their intersection is empty by definition.
Quick review
Practice from earlier coursework, pulled in for this topic.
From Discrete Mathematics I — Logic, Proof, and Structure
What is A times B?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Why
Cartesian product pairs every element of A with every element of B, keeping first and second coordinates in order.
From Discrete Mathematics I — Logic, Proof, and Structure
If |A| = 3 and |B| = 7, what is |A times B|?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Why
The size of a Cartesian product is the product of the two sizes: 3 times 7 = 21.
Correct answer
Also accepted:
Why