Conversion of statements to math

• Mr Davis 97
In summary, to convert a statement into a mathematical expression, one must identify the key mathematical operations and symbols and use them to create an equivalent expression. Common symbols used include addition, subtraction, multiplication, division, equals, and parentheses. Words and phrases in a statement are replaced with appropriate mathematical symbols to accurately represent their meaning. An example of converting a statement to math is "n + 5 = 2n" for the statement "The sum of a number and 5 is equal to twice the number." It is important to be able to convert statements to math in order to solve problems, make predictions, and communicate information in a concise and universal way, and it is a valuable skill in various fields.
Mr Davis 97

Homework Statement

1) 2 is the smallest prime number
2) The area of any bounded plane region is bisected by some line parallel to the x-axis.
3) All that glitters is not gold

The Attempt at a Solution

1) ##\forall p \in P ~~~ (2 \le p)## (where ##p## denotes the set of prime numbers)

2) Let ##r = \text{any bounded plane region}##, ##l = \text{any line parallel to the x-axis}##.
##\forall r \exists l ~~~ (l ~\text{bisects}~ r)##

3) Let ##g = \text{any earthly object}##
##\exists g ~~~ (\text{g glitters} \wedge \neg (\text{g is gold}))##

Is this at all right? I just kind of winged it, and I am assuming that there are a lot of better ways that these could be done.

(3) is fine

(1) says that 2 is a lower bound for the prime numbers, but not that 2 is prime. You need to add a conjunct to specify that 2 is prime.

For (2) I imagine they want you to include the properties 'is a bounded plane region' and 'is parallel to the x axis' in the mathematical statement, rather than in the 'let' part. The verbal version of what I imagine is sought is something like:

For all r, if r is a bounded plane region then there exists l such that l is a line and l is parallel to the x-axis and l bisects the area of r.

For 1) do you mean that I should write ##\forall p \in P ~~~ (2 \le p) \wedge 2 ~~ \text{is a prime number}##

For 2) ##\forall r ~~~ (r ~~ \text{is bounded by a plane region} \rightarrow \exists l ~~~\text{l is a line} ~~ \wedge ~~ \text{l is parallel to the x-axis} ~~ \wedge ~~ \text{l bisects the area of r})##. Is this the best that we can do given that there isn't a lot of notation for things like "parallel to" and "bisects?"

For (1) you can write the second conjunct more concisely as ##2\in P##, given that you defined ##P## as being the set of prime numbers.

(2) looks fine. There is no exact answer here because, to be exact, they'd need to tell us exactly what definitions of predicates and functions we should assume are available to us. Given they have not done that, we are free to assume the existence of any reasonable-sounding predicates we like, such as '#1 is parallel to the x axis' (a unary predicate) and '#1 bisects the area of #2' (a binary predicate).

I'd guess the general idea is to get the student accustomed to writing natural language sentences in logical form, using operators for things like conjunction, entailment and quantification, and employing predicates and functions where required.

Mr Davis 97

1. How do you convert a statement into a mathematical expression?

To convert a statement into a mathematical expression, identify the key mathematical operations and symbols in the statement, and then use those to create an equivalent mathematical expression. It is important to properly understand the meaning and context of the statement in order to accurately convert it into a mathematical expression.

2. What are some common mathematical symbols used in conversion of statements to math?

Some common mathematical symbols used in conversion of statements to math include addition (+), subtraction (-), multiplication (x or *), division (/), equals (=), and parentheses ( ). These symbols are used to represent the different mathematical operations and to group numbers and variables together.

3. How do you handle words or phrases in a statement when converting to math?

When converting a statement to math, words or phrases are replaced with mathematical symbols or operations that represent their meaning. For example, the word "of" is often replaced with multiplication, and the word "more than" can be represented by the greater than symbol (>). It is important to carefully consider the meaning of words and phrases in a statement and use the appropriate mathematical symbols to accurately represent them in the math expression.

4. Can you provide an example of converting a statement to math?

Sure, here's an example: "The sum of a number and 5 is equal to twice the number." This statement can be converted to the following mathematical expression: n + 5 = 2n, where n represents the unknown number. The words "sum," "equal to," and "twice" were replaced with the addition, equals, and multiplication symbols respectively.

5. Why is it important to be able to convert statements to math?

Converting statements to math is important because it allows us to translate real-life situations or problems into mathematical equations or expressions. This can help us better understand and solve problems, make predictions, and communicate information in a more concise and universal way. It is also a necessary skill in many fields such as science, engineering, and finance.

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