# Thread: is this expression a tautology?

1. ## is this expression a tautology?

Show that this is a tautology without creating a truth table (changing logical expression)

(p v q) ^ (-p v r) implies (q v r)

2. anyone? to prove that this is a tautology then (q v r) must be true right? but how do I proof this?

3. $\displaystyle \left( {p \vee q} \right) \equiv \left( { \sim q \to p} \right)$.
$\displaystyle \left( { \sim p \vee r} \right) \equiv (p \to r)$

Thus from the given:
$\displaystyle \left( {p \vee q} \right) \wedge \left( { \sim p \vee r} \right) \equiv \left( { \sim q \to p} \right) \wedge \left( {p \to r} \right) \equiv \left( { \sim q \to r} \right) \equiv \left( {q \vee r} \right)$

4. what I am confused here is that how do you proof:

(~q -> p) ^ (p -> r) is equivalent to (~q-> r)

5. That is a hypothetical syllogism.
Why were you given these problems to do if you don’t know the basics?

6. Hello, EquinoX!

I found a proof . . . but there must be a better way . . .

Show that this is a tautology without creating a truth table:
. . $\displaystyle (p \lor q) \land (\sim p \lor r) \rightarrow (q \lor r)$
Recall that: .$\displaystyle u \rightarrow v \;\;\equiv\;\;\sim u \lor v$ . (def. implication)

Hence: .$\displaystyle \begin{array}{cccccc}p \lor q & \equiv & \sim p \rightarrow q & & &\\ \sim p \lor r & \equiv & p \rightarrow r & \equiv & \sim r \rightarrow\,\sim p & \text{(contrapositve)}\end{array}$

The left side becomes: .$\displaystyle (\sim r \rightarrow\,\sim p) \land (\sim p \rightarrow q)\;\;\equiv\;\;\sim r \rightarrow q\;\;\;\text{(syllogism)}$

Then: .$\displaystyle \sim r \rightarrow q \;\;\equiv\;\;q \lor r$ . (def. implication)

Ha! . . . Plato beat me to it!
{"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": 1, "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, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9423969388008118, "perplexity": 1558.7937530150155}, "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": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945232.48/warc/CC-MAIN-20180421145218-20180421165218-00218.warc.gz"}