Intro to Analysis in-class_Part_2

Intro to Analysis in-class_Part_2 - 0.2 Logic and inference...

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0.2 Logic and inference 9 A = B means that whenever A is true, B must also be true, i.e., it CANNOT be the case that A is true B is false: ( A = B ) ≡ ¬ ( A and ¬ B ) . This means that the truth table for = can be found: A B ¬ B A and ( ¬ B ) ¬ ( A and ¬ B ) T T F F T T F T T F F T F F T F F T F T = A B A = B T T T T F F F T T F F T A B ¬ A ¬ B A = B ¬ ( A and ¬ B ) ¬ A or B ¬ B = ⇒ ¬ A B = A ¬ A = ⇒ ¬ B T T F F T T T T T T T F F T F F F F T T F T T F T T T T F F F F T T T T T T T T If A = B and B = A , then the statements are equivalent and we write “ A if and only if B ” as A ⇐⇒ B,A B, or A iﬀ B. This is often used in deﬁnitions . A B A = B B = A ( A = B ) and ( B = A ) A ⇐⇒ B T T T T T T T F F T F F F T T F F F F F T T T T If you know that A ⇐⇒ B , then you can replace A with B (or v.v.) wherever it appears. A B is like “=” for logical statements. One last rule (DeMorgan):

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Intro to Analysis in-class_Part_2 - 0.2 Logic and inference...

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