About the closed sentences and open sentences in math
OK. So, what is a "sentence"? Well, just like a language sentence, in mathematics a sentence says something. Examples:
1. English language:
- This book is big.
- This ruler is small.
2. Bulgarian language:
- Аз обичам да рисувам.
- Реката е дълбока.
3. Mathematics:
- 7 - 6 = 1
- 8 is an even number.
Alright, so what are these closed and open sentences in math? Well, when it's "closed" it's
always true or always false, and when it's "open" it is
not known if it's true or false. Some examples:
- "10 is an even number." is closed (it is always true).
- "3 is an even number." is closed too (but it is always false).
- "n is an even number." is open (it could be true or false, depending on the value of n; for example, in case "n = 2" -- true; "n = 13" -- false, and so on. But we didn't say what value n has. So, it's an open sentence.)
I hope you understood it. If you need some more examples, here we go:
Closed sentences
- "A rectangle has four corners." (Always true.)
- "182733648 is less than 22." (Always false.)
- "−400 is a negative number." (Always true.)
- "A triangle has ten corners." (Always false.)
Open sentences
- "A square has n sides." Can be true or false (depending on the value of n).
- "z is a negative number." Can be true or false (depending on the value of z).
- "6y = 2x + 2" Can be true or false (depending on the values of x and y).
- "a - b = c - d" Can be true or false (depending on the values of a, b, c and d).
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