Sometimes it is difficult to determine the truth value of a given sentence even when the sentence shows the complete meaning, the sentence may be true for a number of population and false for the same.
Open sentence in Math usually refers to the equation or equality and described as open in the sense. The open sentence math is a general expression that contains one or more variables. There are two types of the sentences those are closed sentences and open sentences.
A closed sentence always has a true value or false value while an open sentence is open when it is not known if it is true or false. Such as,
9 is an odd number [This is closed sentence and always true],
5 is an even number [this is a closed sentence and always falls],
‘n’ is an odd number [this is an open sentence or it could be false or true that depends upon the value of ‘n’],
57 is an odd number [this is closed sentence and is always true],
88 is an even number [this is closed sentence and always true],
In the open sentences (the third example) if n = 3 then it would be true and if n = 4 then the sentence would be false.
Some examples of open sentences are,
3y – 9 = 21, and the only solution for ‘x’ is 10.
4y + 3 > 9, the solutions for x are all Numbers which are greater than 3/2.
x + y = 0, the solution for x and y are all pairs of numbers that are additive.
2x2 – 7x + 6 = 0 [the solution will depend on the value of the variable ‘x’]
An open sentence would either be true or false always but depends upon what values are used. Like,
Triangle has ‘n’ sides [this could be true or false both but depends upon the value of ‘n’]
‘Z’ is a negative number [this would be simplified if the value of ‘z’ is known]
3x = 4y + 2 [could be true or false that will depend upon the value of ‘x’ and ‘y’ totally],
a + b = c + d [this sentence would either true or false and definitely will depend upon the values of a, b, c, d].
As we all know from the property of open sentence, it may be true or false depends on variable value. Here we can easily see that in the equation 3y – 9 = 21, ‘y’ is a variable where the equation depends. So we put the value of ‘y’ in the equation,
For y = 1,
3 * 1 – 9 = -6,
So for y = 1 condition not fulfill. Now we put the value of y = 2,
For y = 2
3 * 2 – 9 = - 3,
Again condition does not full fills. So we again put the value of y = 3,
For y = 3,
3 * 3 – 9 = 0,
After trying lots of positive integers value we put the value y = 10
For y = 10,
3 * 10 – 9 = 21
Here we can easily see that equation fulfill the requirement at y = 10. So the equation is true at y = 10 in the open sentence.
As we all know from the property of open sentence that it should be true or false depend on variable value. Here we can easily see from the inequality 2 + 4x < 6 that x is a variable where we have to put some Integer’s value in it. To check for what value the inequality 2 + 4x < 6 satisfy the condition. So from the inequality
2 + 4x < 6
Now we put the value of x = 1 in the left hand side then 2 + 4 * 1 = 6.
But it is not greater less than 6 so it is not satisfy the condition. And for all values of x which are greater than 1 it also not satisfy the condition. Now if we put value less than 1 means (-) negative integers like -1 then
2 + (-1) * 4 = 2 – 4 = - 2,
In that case we can say that – 2 is less than 6 and for all values ‘x’ which are less than 0 then equation satisfy the condition.
As we all know from the property of Open Sentence, for the values of ‘z’ it may be true or false. Here we can clearly see from the inequality 3z – 2 < 5 that z is a variable where we have to put some integers values and find out that for which values of ‘z' it satisfy the condition.
Now we put the value z = 1 in the left hand side term 3 * 1 – 2 = 3 – 2 = 1,
So for the value z= 1 it satisfy the condition which is true. Now we again put the value in ‘z’, if we put the value z = 2 then in that case,
3 * 2 - 2 = 4,
4 is less than 5 which is true statement so it satisfies the condition. Now we put the value z = 3 then
3 * 3 – 2 = 9 – 4 = 5,
In that case it does not fulfill the condition because 5 are equals to 5 but not less than 5. So it is not a true statement. Now it very clear that for all value of ‘z’ which are greater than 1 it does not fulfill the condition and for all values of ‘z’ which are less than 1 it satisfy the condition.
Here we can clearly see that we have an equation 5n – 3 = 17. Where ‘n’ is a variable in which we put the values of integers. So now we put the value n = 1 n the equation and check whether it satisfy the condition or not.
For n = 1 5 * 1 – 3 = 5 – 3 = 2,
The value 2 does not satisfy the condition. So now we again put the value of n which is 2
For n = 2 5 * 2 – 3 = 10 – 3 = 7,
In that case also it does not satisfy the condition. This time we take the value n = 3. So
For n = 3 5 * 3 – 3 = 15 – 3 =12.
12 also not fulfill the condition. At last we take the value n = 4,
For n = 4 5 * 4 – 3 = 20 – 3 = 17
Here we can clearly see that the value 17 fulfill the right hand side condition. So we can say that for the value of z = 4 the given equation satisfy the condition.
Here we can clearly see that we have an inequality 5 k – 5 > 5 for which we have to find out the solutions where it satisfies the condition and ‘k’ is a variable term. So we put the values of ‘k’ in the left hand side terms and check the condition. In starting we put the value of k = 1 then
5 * 1 – 5 = 5 – 5 = 0,
0 is less than 5 which does not fulfill the condition. So we again put the value k = 2 so,
5 * 2 – 5 = 10 – 5 = 5,
Here we can clearly see that k = 2 we get the value 5 which is equals to 5 but not greater than 5. So condition does not fulfill. We again put the value k = 3 so then in that case
5 * 3 – 5 = 15 – 5 = 10,
Condition fulfill the requirement because at k = 3 then we get value 10 which is greater than 5.