course mth 151
If your solution to stated problem does not match the given solution, you should self-critique per instructions at
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
013. Negation
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
Self-critique Rating:
Question: **** `q001. There are 4 questions in this set.
Two statements are said to be negations of one another if exactly one of the statements must be true. This means that if one statement is true the other must be false, and if one statement is false the other must be true. What statement is the negation of the statement 'all men are over six feet tall'?
*********************************************
Your solution:
Some men are over six feet tall.
Confidence Assessment:
*********************************************
Given Solution:
You might think that the negation would be 'no men are over six feet tall'. However, the negation is in fact 'some men are not over 6 feet tall'.
The negation of a statement, in addition to being false whenever the statement is true, has to include every possibility except those covered by the statement itself. With respect to men being over six feet tall, there are three possibilities:
1. All men are over six feet tall,
2. no men are over six feet tall, and
3. some men are over six feet tall while others aren't.
It should be clear that statements 1 and 2 do not cover the possibility of the third. In fact no two of these statements cover the possibility of the remaining one.
However the following two statements do cover all possibilities:
All men are over six feet tall (the original statement), and
some men are not over six feet tall.
The second statement might seem to be identical to statement 3, 'some men are over six feet tall while others aren't', but it is not. The statement 'some men are not over six feet tall' does not address whether there are men over six feet tall or not, while statement 3 states that there are.
And the statement 'some men are not over six feet tall' might seem to leave out the possibility of statement 2, 'no men are over six feet tall', but again it doesn't address whether or not there are also men over six the tall.
Therefore the negation of the statement 'all men are over six feet tall' is 'some men are not over six feet tall'.
It doesn't matter what's true and what isn't. If the question was to write the negation of 'all men are under 20 feet tall' you would still state the negation as 'some men are under 20 feet tall'. In this case the negation is true, which proves that the statement itself is false.
In the given problem the negation 'some men are under 6 ft tall' is true, proving that the original statement 'all men are over 6 ft tall' is false.
These examples demonstrate why it is important to figure out the negation before you even thing about which statement is true. Either the statement or its negation will be true, but never both.
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
So what is the right answer?
The line that reads
'Therefore the negation of the statement 'all men are over six feet tall' is 'some men are not over six feet tall'.'
indicates the correct answer.
Self-critique Rating:
Question: **** `q002. What is the negation of the statement 'some men are over six feet tall' ?
*********************************************
Your solution:
All men are over six feet tall.
Confidence Assessment:
*********************************************
Given Solution:
While it might seem that the negation of this statement is 'some men are not over six feet tall', the correct negation is 'no men are over six feet tall'. This is because there is an 'overlap' between 'some men are over six feet tall' and 'some men are not over six feet tall' because both statements are true if some men are over six feet while some are under six feet. Negations have to be exact opposites--if one statement is true the other must be false--in addition to the condition that the two statements cover every possible occurance.
Again we have the three possibilities,
1. All men are over six feet tall,
2. no men are over six feet tall, and
3. some men are over six feet tall while others aren't.
The statement ' some men are over six feet tall' is consistent with statements 1 and 3, because if all men are over six feet tall then certainly some men are over 6 feet tall, and if some men are over 6 feet tall and others aren't, it is certainly true that some men are over six feet tall.
The only statement not consistent with 'some men are over six feet tall' is Statement 2, 'No men are over six feet tall'. Thus this statement is the negation we are looking for.
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
I understand
Self-critique Rating:
Question: **** `q003. As seen in the preceding two questions, the negation of a statement that says 'all are' or 'all do' is 'some aren't' or 'some don't', and the negation of a statement that says 'some are' or 'some do' is 'all aren't' or 'none are', or 'all do not' or 'none do'. Each of the following statements can be expressed as and 'all' statement or a 'some' statement. Identify which is which and give the negation of each statement:
1. Every dog has its day. All statement. At least one dog does not have its day.
2. Some roses are black. Some statement. Every rose is not black.
3. Every attempt fails. All statement. Some attempts fail.
4. In some cases the desired outcome isn't attained. Some statement. In all cases the desired outcome is attained.
*********************************************
Your solution:
Confidence Assessment:
*********************************************
Given Solution:
Statement 1 can be expressed as 'All dogs do have their day', a form of 'all do'. The negation of 'all do' is 'some don't'. In this case the negation might be expressed as 'some dogs do not have their day'.
Statement 2 is a straightforward 'some are' statement having negation 'all are not', expressed in this case as 'no roses are black', or equivalently 'there are no black roses'.
Statement 3 can be restated equivalently in 'all do' form as 'all attempts do fail', and is negated in 'some don't' form as 'some attempts do not fail', or equivalently as 'some attempts succeed'.
Statement 4 can be equivalently expressed in 'some are' form as 'some outcomes are not as desired'. This statement is negated by the 'none are' form as 'no outcomes are not as desired', which can then be expressed as 'all outcomes are as desired'.
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
Is this based on universal and existential qualifiers?
Yes. 'all' is univeral, 'some' indicates existential.
Self-critique Rating:
Question: **** `q004. Negate the following statements:
1. No roses are black. Some roses are not black.
2. Some roses are not black. Every rose is back.
3. There were Dodo birds that weren't stupid. All Dodo birds are stupid.
4. There were never turtles that weren't slow. All turtles are slow.
*********************************************
Your solution:
Confidence Assessment:
*********************************************
Given Solution:
Statement 1 says that there is no such thing as a rose which is not black, which says that all roses fail to be black. The negation of 'all are' is 'some aren't', so the negation of 'all roses are not black' is 'some roses are not not black', which is the same as 'some roses are black'.
Statement 2 is a 'some are' statement, negated in the 'all are not' form by 'all roses are not not black', or equivalently, 'all roses are black'.
Statement 3 is equivalent to saying that 'some Dodos birds were not stupid', negated as 'all are not' in the form 'all Dodo birds were not not stupid', or equivalently as 'all Dodo birds were stupid'.
Statement 4 is equivalent of saying that 'all turtles were slow', equivalent of the 'all are' form. This is negated in 'some are not' form by 'some turtles were not slow'.
In the turtle statement I put all instead of some because by reading the statement I couldn? tell if it was a all or some statement but I can see now why it would be an all statement.
&#See my notes and let me know if you have questions. &#