Intereting Posts

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Showing that $\mathbb{R}$ is connected
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The cardinality of a language over a set of variables $V$ with $\#V= κ$
Given several integrals calculate $\int\limits_5^6 f(x) dx$
Can every positive real be written as the sum of a subsequence of dot dot dot
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Prove that $h'(t)=\int_{a}^{b}\frac{\partial\phi}{\partial t}ds$.
Cat Dog problem using integration
How common is the use of the term “primitive” to mean “antiderivative”?
The Duality Functor in Linear Algebra
Computing the order of elements in Dihedral Groups
Height one prime ideal of arithmetical rank greater than 1
The p-adic numbers as an ordered group

We were ask to prove that $|A \cup B| = |A| + |B| – |A \cap B|$.

It was easy to prove it using a Venn diagram, but I think we might

be expected to do if more formally. Is there a formal way?

- Simple Double Summation
- Example of set which contains itself
- Do we have always $f(A \cap B) = f(A) \cap f(B)$?
- Complement of a set and inverse image.
- Compute the following sum $ \sum_{i=0}^{n} \binom{n}{i}(i+1)^{i-1}(n - i + 1) ^ {n - i - 1}$?
- The cross product of two sets
- Is my understanding of antisymmetric and symmetric relations correct?
- Proving $\kappa^{\lambda} = |\{X: X \subseteq \kappa, |X|=\lambda\}|$
- Eventually constant variable assignments
- Finding the coefficient using the multinomial theorem?

$A\cup B = (A\setminus B) \cup (B\setminus A) \cup (A\cap B)$. These three sets are disjoint, so

$$

|A\cup B| = |A\setminus B| + |B\setminus A| + |(A\cap B)|

$$

But $A\setminus B = A\setminus(A\cap B)$, so $|A\setminus B|=|A|-|A\cap B|$. A similar equality holds for $|B \setminus A|$. Substitution of these into the displayed equation above yields your result.

Of course, one might need to formally show that $|A\setminus (A\cap B)| = |A|-|A\cap B|$. I can’t decide if this is any less obvious than the original proposition…

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- Dirac delta distribution & integration against locally integrable function
- Cauchy Schwarz inequality for random vectors
- The blow up of of the plane and the Moebius band
- If $f(a) = g(a)$ and $f'(x) < g'(x)$ for all $x \in (a,b)$, then $f(b) < g(b)$
- Expected number of steps to walk through points by multiple walkers
- Catalan number interpretation
- Is it possible to build a circle with quadratic Bézier curves?
- Rudin assumes $(x^a)^b=x^{ab}$(for real $a$ and $b$) without proof?
- Closed model categories in the sense of Quillen vs the modern sense
- How to tell if a directed graph is acyclic from the adjacency matrix
- Computing A Multivariable Limit: $\lim_{(x,y) \to (0,0)}\frac{2x^2y}{x^4 + y^2}.$
- Yoneda's lemma and $K$-theory.