013 Negation

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course Mth 151

3/13 12

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/file

3_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

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Self-critique (if necessary):

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Self-critique Rating:

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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'?

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Your solution:

Some men are not over six feet tall.

confidence rating #$&*: 3

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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'.

Think of it this way:

If I claim that all men are over six feet tall, all you need to prove me wrong is

one man who isn't.

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.

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Self-critique (if necessary):OK

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Self-critique Rating:3

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Question: `q002. What is the negation of the statement 'some men are over six feet

tall' ?

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Your solution:

No men are over six feet tall.

confidence rating #$&*: 3

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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.

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Self-critique (if necessary):OK

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Self-critique Rating:3

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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.

2. Some roses are black.

3. Every attempt fails.

4. In some cases the desired outcome isn't attained.

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Your solution:

Some dogs do not have their day.

No roses are black.

Some attempts do fail.

All outcomes are as desired.

confidence rating #$&*: 3

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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'.

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Self-critique (if necessary):OK

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Self-critique Rating:3

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Question: `q004. Negate the following statements:

1. No roses are black.

2. Some roses are not black.

3. There were Dodo birds that weren't stupid.

4. There were never turtles that weren't slow.

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Your solution:

Some roses are not black.

All roses are black.

All Dodo birds were not stupid.

Some turtles were not slow.

confidence rating #$&*: 3

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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'.

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Self-critique (if necessary):

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Self-critique rating:

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Question: `q004. Negate the following statements:

1. No roses are black.

2. Some roses are not black.

3. There were Dodo birds that weren't stupid.

4. There were never turtles that weren't slow.

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Your solution:

Some roses are not black.

All roses are black.

All Dodo birds were not stupid.

Some turtles were not slow.

confidence rating #$&*: 3

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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'.

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Self-critique (if necessary):

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Self-critique rating:

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