Show that if $(x_1,x_2)$ is defined to be $\{\{x_1\},\{x_1,x_2\}\}$ then $(x_1,x_2)=(y_1,y_2)$ iff $x_1=y_1$ and $x_2=y_2$
22482
How can an ordered pair be expressed as a set?
151805
Prove that $\tau (x_1 x_2 ... x_k) \tau^{-1} = (\tau(x_1) \tau(x_2) ... \tau (x_k))$
124467
Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k[X_1,X_2,X_3,X_4]$?
5347
Using the definition of a vector space to prove that $0x = 0$ and $(-1)x = -x$
959
Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.
155228
$X_i, X_j$ independent when $i≠j$, but $X_1, X_2, X_3$ dependent?
97801
If $R_{1}$ and $R_{2}$ are relations on a set S with $R_{1};R_{2}=R_{2};R_{1}$. Then $R_{1}$ and $R_{2}$ are bijective maps
62077
Prove that if $W_1\subseteq V$ finite-dimensional, then there is $W_2\subseteq V$ such that $V=W_1\oplus W_2$
53660
Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q  \implies  \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $
67949
$S_3$ permutes the elements of $\mathbb{R}^3$, show that $\sigma (x_1, x_2, x_3) = ( x_{\sigma^{-1}(1)}, x_{\sigma^{-1}(2)}, x_{\sigma^{-1}(3)})$.
21891
If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t
50012
$f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
119782
Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V.
37985
Show that $|x|^2 = |y|^2$ with $A$ orthogonal and $Ax = \cos (θ)x − \sin (θ)y$, $Ay = \sin (θ)x + \cos (θ) y$
43310
Proving $A\cap(B-C)$ is equal to $(A\cap B)-(A\cap C)$
39908
Finding the error in this proof that 1=2
92291
$U_1, U_2, ..., U_k$ subspaces of linear space $X$ with same dimensions, prove $X=U_1\oplus V=U_2\oplus V=...=U_k\oplus V$