Prove that $P(A ∩ B) ≤ P(A ∪ B) ≤ P(A) + P(B)$

I don´t know if my proof is right. I separated it into 2:

$$1) P(A ∩ B) ≤ P(A ∪ B)$$

Pf: $$P(A ∪ B) = P(A\setminus B) + P(B\setminus A) + P(A ∩ B)$$ and we know that $$0 ≤ P(A\setminus B) ≤ 1$$ and $$0 ≤ P(B\setminus A) ≤ 1$$ so $$P(A ∩ B) ≤ P(A ∪ B)$$

$$2) P(A ∪ B) ≤ P(A) + P(B)$$

Pf: $$P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$$ and we know that $$0 ≤ P(A ∩ B) ≤ 1$$ so $$P(A ∪ B) ≤ P(A) + P(B)$$

Therefore, $$P(A ∩ B) ≤ P(A ∪ B) ≤ P(A) + P(B)$$

• What does $P(X)$ mean? The power set of $X$? The probability of an event $X$? What does P(X) \le P(Y)$mean? – fleablood Jan 26 at 17:51 • @fleablood from context,$P(X)$is clearly referring to probability and$\le$means the usual less than or equal relation on real numbers. @ Regina, you have some misused symbols. Set difference is \ , not to be confused with / which is more often for division or quotient groups etc... Visit this page for information on how to type with MathJax and$\LaTeX\$ on this site. – JMoravitz Jan 26 at 18:08
• If you use Venn diagram the problem will simplify and you will be able to visualize better – SNEHIL SANYAL Jan 27 at 4:08

Looks fine. In the first proof you are really making use of the fact that $$P(A\setminus B)\geq 0$$ and $$P(B\setminus A)\geq 0$$. In the second proof you are making use of the fact that $$P(A\cap B)\geq 0$$. You aren't really making use of the fact that the probabilities are at most one so I would omit it from the explanation.
{"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": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 13, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://math.stackexchange.com/questions/3088519/prove-that-pa-%E2%88%A9-b-%E2%89%A4-pa-%E2%88%AA-b-%E2%89%A4-pa-pb", "fetch_time": 1563402084000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2019-30/segments/1563195525414.52/warc/CC-MAIN-20190717221901-20190718003901-00105.warc.gz", "warc_record_offset": 456328853, "warc_record_length": 36076, "token_count": 534, "char_count": 1488, "score": 3.859375, "int_score": 4, "crawl": "CC-MAIN-2019-30", "snapshot_type": "latest", "language": "en", "language_score": 0.8087345361709595, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00035-of-00064.parquet"}