MandysNotes

Sunday, 06 April 2014 00:00

Truth Values for Conjunction

By  Gideon
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Suppose that your firend is thinking of an animal, and you guess that he is thinking of a platypus. 

You say:

"The animal is both duck-billed and a mammal."

In other words, you are saying that \( A \cap B\) is true. 

You are saying that the animal will be in the green section of the figure below.

AandBtrue

 

 

 If your friend tells you that the animal he is thinking of is duck-billed, and is a mammal, then he is telling you that \(A\) is true, and that \(B\) is true, and hence that your guess was right, that \(A \cap B\) is true. 

On the other hand, if your friend tells you that the animal is duck-billed, but is not a mammal, then he is telling you that \(A\) is true, but that \(B\) is false. 

We can represent this by the following figure:

AtrueBfalse

 Because the animal that your friend is thinking of will with certainty not be in the red area, \( A \cap B \) is clearly false. 

Similarly, if he tells you that the animal is not duck-biled, but is a mammal, he is telling you that \(A\) is false, but that \(B\) is true. 

We represent this as:

AfalseBtrue

Once again we see that \( A \cap B\) is in the red area of this diagram, and so must be false. 

Finally if your friend tells you that the animal is not duck-billed and not a mammal, then he is telling you that \(A\) is false, and that \(B\) is false. 

AfalseBfalse

 

In this case, \(A \cap B\) is clearly false. 

We can represent all these truth values in a truth table for \(A \cap B.\)

 AandBtruthtable1

 Note that \(A \cap B\) is only true when both \(A\) and \(B\) are true, otherwise it is false. 

 

 

 

 

Read 1408 times Last modified on Sunday, 06 April 2014 18:37
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