cmi-entrance 2016 Q1

cmi-entrance · India · pgmath 4 marks Number Theory Combinatorial Number Theory and Counting
We say that two subsets $X$ and $Y$ of $\mathbb{R}$ are order-isomorphic if there is a bijective map $\phi : X \longrightarrow Y$ such that for every $x_1 \leq x_2 \in X$, $\phi(x_1) \leq \phi(x_2)$, where '$\leq$' denotes the usual order on $\mathbb{R}$. Choose the correct statement(s) from below:
(A) $\mathbb{N}$ and $\mathbb{Z}$ are not order-isomorphic;
(B) $\mathbb{N}$ and $\mathbb{Q}$ are order-isomorphic;
(C) $\mathbb{Z}$ and $\mathbb{Q}$ are order-isomorphic;
(D) The sets $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ are pairwise non-order-isomorphic.
We say that two subsets $X$ and $Y$ of $\mathbb{R}$ are order-isomorphic if there is a bijective map $\phi : X \longrightarrow Y$ such that for every $x_1 \leq x_2 \in X$, $\phi(x_1) \leq \phi(x_2)$, where '$\leq$' denotes the usual order on $\mathbb{R}$. Choose the correct statement(s) from below:\\
(A) $\mathbb{N}$ and $\mathbb{Z}$ are not order-isomorphic;\\
(B) $\mathbb{N}$ and $\mathbb{Q}$ are order-isomorphic;\\
(C) $\mathbb{Z}$ and $\mathbb{Q}$ are order-isomorphic;\\
(D) The sets $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ are pairwise non-order-isomorphic.