Prove that $\phi$ is a partial relation for. Let $\phi$ is a relation on $\mathbb{R}\times\mathbb{R}$, $(a,b)\phi (x,y) \iff a\leq x$ and $b\leq y$
55328
Let $σ$ be the relation on $R×R$ in which $(a,b) σ (x,y)$ if and only if $a ≤ x$ and $b ≤ y$. Prove that $σ$ is a partial order relation.
61103
Prove that $\mathbb{|Q| = |Q\times Q|}$
83291
Prove that $p(x)=(x-a)q(x)$ for some $q(x)\in R[x]$ of degree= (degree of $p)-1$.
70552
Prove that $\mathcal{P}(A)⊆ \mathcal{P}(B)$ if and only if $A⊆B$.
21807
on proving bac-cab rule, $\vec{A} \times ( \vec{B} \times \vec{C})= \vec{B} (\vec{A} \cdot\vec{C})- \vec{C}(\vec{A}\cdot\vec{B})$
50273
How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?
25155
Suppose $R$ and $S$ are transitive relations on $A$. Prove that if $S \circ R \subseteq R \circ S$ then $R \circ S$ is transitive.
57269
A special bijection between $\mathbb{R}\times \mathbb{R}$ and $\mathbb{R}$
59424
Showing that $B=\{(x,y) \in \Bbb{R^2} \mid \ xy>0 \}$ is an open set
39455
Prove that $(X\times Y)\setminus (A\times B)$ is connected
138819
Prove that $\phi(n) \geq \sqrt{n}/2$
120971
(Revised) Prove that $\phi$ is a group homomorphism and find the kernel.
45179
When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
132487
Proving that if $V$ is an inner product space and $U_1, U_2$ are subsets, then $U_1 \subset U_2$ if and only if $U^{\perp}_2 \subset U^{\perp}_1$
34768
Proving $(p \oplus q) \oplus r=p \oplus (q \oplus r)$
116215
Prove that $P(A \cap B\mid C)=P(A \mid (B \cap C))P(B\mid C)$
26658
A proof for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$.