https://www.youtube.com/watch?v=RRcRDVCA7Bg Even functions $f$ are symmetric across the $y$-axis, meaning $f(-x)=f(x)$. Odd functions $g$ are symmetric across the origin, meaning $g(-x)=-g(x)$. As long as the domain of a function is symmetric across $0$ on the real line, it turns out that the function can be uniquely written as a sum of an even function and […]
Unique Even and Odd Parts of a Function Read More »
Adding a number to its negative produces $0$. Multiplying a non-zero number by its reciprocal produces $1$. We show that, given a fixed element that has an inverse in an associative binary operation (like addition or multiplication), the inverse is unique. So a number cannot have more than one negative or more than one reciprocal.
Inverses in a Binary Operation are Unique Read More »
Given a binary operation, an identity element is like $0$ for addition or $1$ for multiplication: operating on a different number with it does not change that number. We show that, if an identity element exists, it is unique. So there cannot be two identity elements.
Identity Element of a Binary Operation is Unique Read More »
https://www.youtube.com/watch?v=oTXvwc3k8ag Cantor’s paradox derives a contradiction from the assumption that there is a set of all sets. Like Russell’s paradox, this proves that there can be no set whose elements are all sets; in other words, the mathematical universe is not a set. The main tool is Cantor’s theorem, a topic on which we have
Cantor’s Paradox in Set Theory Read More »
https://www.youtube.com/watch?v=7FQu540d3-A By repeatedly taking the power set $mathcal{P}(S)$ of an infinite set $S$, Cantor’s theorem shows that these new infinities get strictly “bigger and bigger.” So there exists an infinite hierarchy of infinities in the sense that the lower infinities cannot map surjectively onto the higher infinities (but injective maps are possible).
Cantor’s Theorem in Set Theory Read More »
http://www.youtube.com/watch?v=ZeJZFaP6dL8 A function $f: X to Y$ is bijective (both injective and surjective) if and only if it has an inverse $f^{-1} : Yto X$ (a function that is both a left-inverse and a right-inverse). We prove this biconditional result here.
Bijective Function if and only if Inverse Exists Read More »
https://www.youtube.com/watch?v=Drsqa4XFgJQ Bertrand Russell’s paradox in set theory shook the foundations of the mathematical world in 1901 by showing that naive set theory leads to a contradiction. This caused mathematicians to adopt the convention that one cannot summon elements directly from the entire mathematical universe into a set, and that there is no set of all
Bertrand Russell’s Paradox in Set Theory Read More »
https://www.youtube.com/watch?v=cCEvHZgq188 One of the most powerful and general concepts in math is the idea of a partial order. We show that, given that we are working within a universl set, its subsets fall into a partial order under the subset relation $Xsubseteq Y$. This means that the subset relation is reflexive, antisymmetric, and transitive.
Subset Relation is a Partial Order Read More »
https://www.youtube.com/watch?v=LJjJbljEOVs Logical quantifiers are often bounded like $forall xin S: P(x)$. That is, we say that universally or existentially quantified elements satisfy a predicate (like being a member of some set) before stating the condition to be fulfilled. We describe such logical sentences here.
Bounded Quantifiers in Logic Read More »
https://www.youtube.com/watch?v=DOu1kjAXutU Learn to negate logical expressions involving disjunctions and conjunctions using De Morgan’s laws $$neg(pvee q) equiv neg p wedge neg q,$$ and $$neg(pwedge q) equiv neg p vee neg q.$$
De Morgan’s Laws in Logic Read More »