# Is this quantification correct?

Let $Q(x,y)$ be the statement $y = 2x +1$ what are the values of the following. The universe is $Z^+$ {1,2,3,....}.

(a) $\forall x\exists y Q(x,y)$ This is true

(b) $\exists x \forall y Q(x,y)$ This is false.

-

That's correct. – Myself Mar 9 '11 at 1:38

## 1 Answer

The first is true. You are correct. For the second, truth would imply that 2x+1=1 has a positive integer as a solution, which is false, so you are correct again.

-

It's worse than that for the second one, it would imply that for some $x\in\mathbf Z^+$ it holds that $2x-y = 1 = 2x-y'$ for all $y,y'\in\mathbf Z^+$, so it would imply that $y=y'$, thus $|\mathbf Z^+|=1$. – Myself Mar 9 '11 at 1:49
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 10, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.954059362411499, "perplexity_flag": "head"}