1 Intro to Logic Supplementary Notes Prepared by Raymond Wong Presented by Raymond Wong

2 e.g.1 (Page 12) Show that “ (w  v)  u ” and “ (w  v)  u ” are not equivalent. Consider w = T v = T and u = F Consider (w  v)  u= (T  T)  F = T  F = T Consider (w  v)  u= (T  T)  F = T  F = F

3 e.g.2 (Page 24) E.g., s  t st s  t TTT TFF FTT FFT

4 e.g.2 st s  t TTT TFF FTT FFT s ~ “ Raymond takes course Discrete Mathematics ” t ~ “ Raymond knows the RSA algorithm ” The following statement S is the fact: if “ Raymond takes course Discrete Mathematics ”, then “ Raymond knows the RSA algorithm ”

5 e.g.2 st s  t TTT TFF FTT FFT s ~ “ Raymond takes course Discrete Mathematics ” t ~ “ Raymond knows the RSA algorithm ” The following statement S is the fact: if “ Raymond takes course Discrete Mathematics ”, then “ Raymond knows the RSA algorithm ” Suppose that we know the following. “ Raymond takes course Discrete Mathematics ” “ Raymond knows the RSA algorithm ” The statement S is correct.

6 e.g.2 st s  t TTT TFF FTT FFT s ~ “ Raymond takes course Discrete Mathematics ” t ~ “ Raymond knows the RSA algorithm ” The following statement S is the fact: if “ Raymond takes course Discrete Mathematics ”, then “ Raymond knows the RSA algorithm ” Suppose that we know the following. “ Raymond takes course Discrete Mathematics ” “ Raymond does not know the RSA algorithm ” The statement S is not correct.

7 e.g.2 st s  t TTT TFF FTT FFT s ~ “ Raymond takes course Discrete Mathematics ” t ~ “ Raymond knows the RSA algorithm ” The following statement S is the fact: if “ Raymond takes course Discrete Mathematics ”, then “ Raymond knows the RSA algorithm ” Suppose that we know the following. “ Raymond does not take course Discrete Mathematics ” “ Raymond knows the RSA algorithm ” These two events does not provide any information to check whether the statement S is correct or not. The principle of the Excluded Middle: We don ’ t know whether the statement S is correct or not in this case. A statement is correct exactly when it is not false. ?

8 e.g.2 st s  t TTT TFF FTT FFT s ~ “ Raymond takes course Discrete Mathematics ” t ~ “ Raymond knows the RSA algorithm ” The following statement S is the fact: if “ Raymond takes course Discrete Mathematics ”, then “ Raymond knows the RSA algorithm ” Suppose that we know the following. “ Raymond does not take course Discrete Mathematics ” “ Raymond does not know the RSA algorithm ” These two events does not provide any information to check whether the statement S is correct or not. The principle of the Excluded Middle: We don ’ t know whether the statement S is correct or not in this case. A statement is correct exactly when it is not false. ?

9 e.g.3 (Page 24) E.g., s  t st TT TF FT FF s  tt  ss  t T F T T T T F T T F F T s  t and t  s

10 e.g.4 (New Notes) s  t is equivalent to  s  t st s  t TTT TFF FTT FFT You need to remember this rule. ss t  s  t F F T T T F T F T F T T

11 e.g.5 (New Notes) We want to simplify the following where p is a statement. p  true = p p  false = false p  p = p p   p = false p  true = true p  false = p p  p = p p   p = true