#$&* course MTH 151 03/12/13 5:50 013. Negation
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q002. What is the negation of the statement 'some men are over six feet tall' ? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ 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. 2. Some roses are black. 3. Every attempt fails. 4. In some cases the desired outcome isn't attained. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Some dogs do not have their day All roses are black No attempts fails In all cases the desired outcome is attained confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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'. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I dont think I have this problem down to a t. I need to do more work on the question and work on the correct answers. ------------------------------------------------ Self-critique Rating: OK ********************************************* 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Some dogs do not have their day All roses are black
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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'. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I dont think I have this problem down to a t. I need to do more work on the question and work on the correct answers. ------------------------------------------------ Self-critique Rating: OK #*&! ********************************************* 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Some dogs do not have their day All roses are black
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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'. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I dont think I have this problem down to a t. I need to do more work on the question and work on the correct answers. ------------------------------------------------ Self-critique Rating: OK #*&!#*&!