Monday, April 17, 2017

Venn diagrams and the meaning of English sentences



I was looking some English language materials recently and came across these two sentences. The basic question from the standpoint of understanding English was around the effects of commas and the meaning of similar statements in English. Thinking about it, it occurred to me that the question was about sets and logic as much as English comprehension. The original problem was stated similar to

In a certain math class there are Hardworking students and Lazy students. The class writes two tests. A student can Pass or Fail each test.

These are the overall results of the two tests.

1. The students, who were lazy, failed.

2. The students who were lazy failed.

Questions for students

- draw Venn diagrams with Hardworking, Lazy, Pass, and Fail for each of the two tests
- did any Hardworking students Fail the first test (yes, no, possibly)
- did any Lazy students Pass the first test (yes, no, possibly)
- did any Hardworking students Fail the second test (yes, no, possibly)
- did any Lazy students Pass the first test (yes, no, possibly)
- is it possible that everyone failed the first test?
- is it possible that everyone failed the second test?
- which test was probably harder? explain your reasoning


Now the first bit is English comprehension, as the only difference between the sentences is the punctuation. This part might be contentious so I put in my reading of the sentences. I'd be curious to understand how others may read it.

In the first sentence, my reading is: the punctuation makes laziness a property of those who failed the test.

In the second sentence I read it: the statement applies to the class as a whole.

So given that I can rewrite the sentences in an equivalent form that can be used for Venn diagrams

1 => 1.1 everyone who failed is lazy

2 => 2.2 everyone who is lazy failed

From there it's possible to create Venn diagrams and answer the questions.

I used creatly which was a nice online program. Easy to use and powerful. +1 to creately.

These are the Venn diagrams I came up with

For the first test

For the second test

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