So last time, we saw some basic symbolic logic and an introduction on formal proofs. Today we will expand on that, and also start talking about sets - which will lead us, in the future, to learn some pretty interesting stuff about Set Theory - which many consider to be the theoretical basis for all mathematics and is a personal favorite subject of mine.

Let's start with sets.

What is a set?

Well, a Set is any group of elements. A set of numbers, a set of words, a set of people, of objects, of ideas. Anything. Even a set of sets.

Generally, sets are denoted with curly brackets, i.e {...}. Inside the brackets, we put the elements of the set. For example, S = {1,2,3,4} is a set that has 4 elements, which are the numbers 1, 2, 3 and 4. The order of the elements is meaningless, and so are repetitions. For example, all the following sets are identical:

• {1,2,3,4}
• {3,2,4,1}
• {1,1,2,4,2,1,3,3,1,4}

Almost any group of elements you can think of can be defined as a set. There are certain axioms that dictate what sets can or can't be defined, but more on that in the future. Sets can be elements of other sets, too; for example: {1,2,3,{4,5,6,7},8,9}. This is a set that has 6 elements: the numbers 1, 2, 3, 8 and 9, and the set {4,5,6,7}.

There exists one set which has no elements in it. It is called the "Empty Set", and is denoted: Ø. Sometimes it is also denoted {} (empty curly brackets), but the former notation is more common.

Notations and operations on sets:

There are several notations we should mention before we keep talking about sets.

• Belonging/membership: denoted "∈". x∈A means "x belongs to/is an element of A". Likewise, x∉A means "x doesn't belong to A". For example, if we define A to be the set of all Marvel heroes and B to be the set of all DC heroes, and we define x as Spider-Man, we can say that x∈A and x∉B.
• Containment: denoted "⊆". Containment is very often confused with belonging as they are both very similar concepts. But whereas belonging is a relation between a set and an element, containment is a relation between a set and another set. We say that A⊆B (A is contained in B) if A is a subset of B, i.e every element of A is also an element of B. For example, {1,2,3}⊆{1,2,3,4}. As another example, if we define A to be the set of Marvel characters and B to be the set of Marvel villains, then we can say that B⊆A. Do note that if set A is an element of set B, for example, B = {1,2,3,A,4,5}, then it is NOT true that A⊆B, but it IS true that A∈B. Interesting note: the Empty Set is contained in every set. Since it has no elements to begin with, then we can, in fact, say that "every element of Ø is also an element of A", no matter what set "A" is. Also, every set is a subset of itself (A⊆A no matter what A is). Note that the notation "⊂" also exists and means "strict containment", i.e A⊂B means that A is a subset of B but is NOTequal to B.
• Union: denoted "∪". The union of two sets A and B (A∪B) is the set that you get when you combine A and B. For example, if we define A = {1,2,3} and B = {4,5,6}, then A∪B = {1,2,3,4,5,6}. Or if A = {1,2,3} and B = {1,4,5} then A∪B = {1,2,3,4,5} (since repetitions are meaningless in sets, we drop the extra '1'). As another example, if A is the set of all decent MCU films and B is the set of all decent DCEU films, then A∪B = A (KIDDING!!! Don't kill me!!!)
• Intersection: denoted "∩". The intersection of two sets A and B (A∩B) is the set of all elements that belong in A and in B both. For example, if we define A = {1,2,3} and B = {1,2,4}, then A∩B = {1,2}. If A = {1,2,3} and B = {4,5,6}, then A∩B = Ø. If A is the set of all things in the world, and B is the set of all things that Jon Snow knows, then A∩B = Ø.
• Relative complement: denoted "\". The relative complement of two sets A and B (A\B) is the set of all elements of A that are not elements of B. For example, if we define A = {1,2,3,4} and B = {1,2,3,5}, then A\B = {4} (note: a set that only has one element is called a "Singleton"). If A⊆B, then A\B = Ø (since every element of A is necessarily an element of B).

There are several more of those, such as the Cartesian Product of sets which we will surely meet sooner or later, but these are the important ones for now. So, why do we need this to continue talking about true, false and formal proofs? Well, that's because without sets, we wouldn't be able to talk about quantifiers and predicates!

So let's talk about quantifiers:

Well, I lied; there's one more thing I wanna say about sets, but it's short, I promise!

There are a few "special" sets that are commonly used in mathematics, and have their own special notations. I'll say a few words about each of them before we proceed:

• Integers: denoted "ℤ" (fancy Z, from the German word "Zahlen" - "numbers". The Power of German Originality). An integer is a number whose fractional part equals zero (i.e has only 0 after the decimal point). Basically, ℤ = {0,1,-1,2,-2,3,-3,...}.
• Natural numbers: denoted "ℕ" (fancy N). Natural numbers are the positive (or non-negative) integers. So basically, ℕ = {0,1,2,3,4,5,...}. Important note: sometimes 0 is considered a natural, sometimes it isn't, it mostly depends on which field of mathematics you're studying. In discrete mathematics and combinatorics for example, 0 is a natural number, and in calculus it isn't. For the purpose of this blog, we'll say that it is.
• Rational numbers: denoted "ℚ" (fancy Q, from the Italian word "Quoziente" - "quotient"). A rational number is any number that can be expressed as the quotient of 2 integers. For example: ½ is a rational number since it is the quotient of the integers 1 and 2.
• Real numbers: Everything else, basically. Denoted "ℝ" (fancy R). The set of real numbers is the set of all numbers, both rational and irrational. For example, π∈ℝ. √2∈ℝ (it can be proven that the square root of any prime number is irrational, maybe in one of the future entries). e∈ℝ (Euler's Constant, we will meet it again in future entries probably). Note that the set of irrationals has no special notation and is simply expressed as ℝ\ℚ (the relative complement of the reals and the rationals, basically all reals that aren't rationals).

Note that: ℕ⊆ℤ⊆ℚ⊆ℝ.

Set-builder notation: a way of defining our own subsets of certain sets which we already know, according to some condition. The set-builder notation is used as follows: A = {x∈[set] | [condition]}. For example, if we write: A = {x∈ℤ | x<0}, that would mean that we defined A to be the set of all negative integers.

Now we can continue so let's talk about quantifiers. There are two quantifiers which are commonly used in mathematics:

• There exists: denoted "∃" (reversed E, as in "Exists").
• For all/for each: denoted "∀" (reversed A, as in "All").

A quantifier comes before a certain variable which belongs to a certain set, followed by a certain proposition regarding the variable. For example, if we write: ∃x∈ℕ(x>5), it means "there exists a natural number "x" which is bigger than 5" (note that "there exists" doesn't necessarily mean just one). As another example, we can write: ∀x∈ℕ(x>-1), and that means "all natural numbers are bigger than -1".

Of course, if we want to negate a quantifier ("not for all" or "there does not exist"), we can use the negation symbol we saw in the previous entry: "¬". However, a more elegant way of negating a quantifier is using the other quantifier and reversing the statement. For example, if we want to say "there does not exist a natural number smaller than -5", we can either write: ¬(∃x∈ℕ(x<-5)), or we can write: ∀x∈ℕ(x>-5). That is a common way of disproving/refuting mathematical statements. Most mathematical statements are in the form of "for all" (for example, "all differentiable functions are continuous"), and in order to refute such a statement we need to find a counterexample - or in other words, show that "there exists a differentiable function which isn't continuous" (there doesn't).

A statement can have more than one quantifier of course. For example: ∀x∈ℕ(∃y∈ℤ(x+y=0)), which means "for all natural numbers "x" there exists an integer "y" such that x+y=0". Also, these kinds of statements can (and usually do) use the symbolic logic we saw in the previous entry (¬ ⇒ ⇔ ^ ∨ ⊕). For example, consider Goldbach's Conjecture: Goldbach's Conjecture suggests that every even integer greater than 2 is the sum of 2 primes. How do we Write it? First, let's define:

• A = {x∈ℕ | (x>2)^((x/2)∈ℤ}
  • In other words: A is the set of all naturals that are larger than 2, which yield no remainder when divided by 2.
• B = {x∈ℕ |(x≥2)^(∀y∈ℕ(y≠1)^(y≠x)⇒(x/y)∉ℤ)}
  • In other words: B is the set of all naturals larger than or equal to 2, that if you divide them by any number other than 1 or themselves, you get a remainder.
• We write: ∀x∈A(∃y,z∈B(y+z=x))
  • In other words: for each x in A, there exist y,z in B, such that y+z=x.
• Side note: Goldbach's conjecture is yet to be proven or refuted to this day.

∀∃ vs. ∃∀:

Yes, there's a difference. Consider the two following two suggestions:

1. For each superhero, there exists a supervillain.
2. There exists a supervillain for all superheroes.

These two sentences are a classic example for the stark difference between ∀∃ (for each... there exists...) and ∃∀ (there exists... for all...).

Consider the first suggestion: "For each superhero, there exists a supervillain." You name a superhero, and I name their villain. You say Superman, I say Lex Luthor. You say Thor, I say Loki. This suggestion implies that for each superhero there exists a villain, but nothing assures us that two heroes would have the same villain.

Now, consider the second: "There exists a supervillain for all superheroes." There's a specific villain you can name who is an enemy to all heroes. You name Thanos, I name the entire Marvel Universe. You name Darkseid, I name the entire DC Universe. You name Tom King, I name the entire comics industry.

Predicates:

Now, on to predicates. A predicate (usually marked as "P") is a proposition on a certain variable which can either be true or false. For example, if we define a predicate P, by P(x) = "x is even". The set A = {x∈ℕ | P(x)} would be the set of all natural numbers for which P is true (in this case, the set of all even naturals). A predicate can also be defined on more than one variable. For example, we can define P(x,y) = "x is smaller than y". This kind of predicate cannot be used for a set-builder notation, however. Let's see how we can express Goldbach's Conjecture which we saw earlier using predicates:

• P(x) = "x is even"
• Q(x) = "x is prime"
• We write: ∀x∈ℕ((x>2)^(P(x)))⇒∃y,z∈ℕ((Q(y))^(Q(z))^(y+z=x))
  • In other words: For each natural number x, if x is larger than 2 and is even, it implies that there exist two naturals y,z such that y is prime and z is prime and y+z=x.

Examples for the use of quantifiers and predicates:

1. All math must die:

• A = the set of all men.
• P(x) = "x must die".
• We write: ∀x∈A(P(x)) - "for all men "x" there holds that x must die".

2. Lord of the Reals:

• A = the set of all living beings.
• B = the set of all rings.
• P(x,y) = "x rules y".
• We write: ∃x∈B(∀y∈A(P(x,y))) - "There exists a ring "x" such that for each living being "y" there holds that x rules y".
  • Note that I used ∃∀ and not ∀∃.

3. Guns N' Quantifiers:

• A = the set of all people.
• a = you.
• P(x,y) = "x needs y".
• We write: (∀x∈A(∃y∈A(P(x,y))))^(∃b∈A((b≠a)^(∃c∈A(P(b,c))))) - "For each person "x" there exists a person "y" such that x needs y, and there exists a person "b" that isn't you, for which exists a person "c" such that b needs c".

*Note: I'm using so many parentheses because that's the most "formal" way of writing such statements, even though it's kinda clustered. Usually, some parentheses are dropped where they aren't really necessary for understanding the statement.

In conclusion:

Well, this was a nice little intro into the fascinating world of mathematics. I wanted to start with those 2 entries, even though they might be a tiny bit dull, because that's really as basic as it gets, and in future entries we will again meet everything we've discussed so far. I promise, it's gonna get more interesting from now on. What we just discussed about sets prepares us for some Set-Theory, and discussions on some very interesting stuff like cardinalities of infinity, and now that we know how to use quantifiers and predicates we can use that to show and prove some pretty mind-blowing theorems. Trust me, there's more to come and it's gonna be fascinating.

Thanks for reading!