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Buku Text Discrete Mathematics and its Applications Kenneth H. Rosen – 6 th edition Mc Graw-Hill International Edition http://arief.ismy.web.id

Conditional Statements “If p then q” “p implies q” “p  q”; p: hypothesis, q: conclusion. Conditional: the truth of statement q is conditioned on the truth of statement p Example: IF 36 is divisible by 6, THEN 36 is divisible by 3 IF you show up for work Monday morning, THEN you will get the job. Under what circumstances is the above sentence false? http://arief.ismy.web.id

Truth Table for Conditional Statements Definition p  q is false when p is true and q is false; otherwise it is true. http://arief.ismy.web.id

Truth Table for p  ~q  ~p http://arief.ismy.web.id

p  q  r  (p  r)  (q  r) OR p  q  r  (p  r)  (q  r) http://arief.ismy.web.id

Representation of IF-THEN as OR p: you do not get to work on time q: you are fired IF you do not get to work on time THEN you are fired ~p: you get to work on time p  q  ~p  q You get to work on time OR you are fired http://arief.ismy.web.id

Representation of IF-THEN as OR Truth table for p  q  ~p  q http://arief.ismy.web.id

De Morgan’s Laws Definition: The negation of an AND statement is logically equivalent to the OR statement in which each component is negated. ~(p  q)  ~p  ~q The negation of an OR statement is logically equivalent to the AND statement in which each component is negated. ~(p  q)  ~p  ~q http://arief.ismy.web.id

The Negation of a Conditional Statement The negation of “IF p THEN q” is logically equivalent to “p and not q” ~(p  q)  p  ~q Show the equivalence by using Morgan Law ~(p  q)  ~(~p  q)  ~(~p)  ~q  p  ~q http://arief.ismy.web.id

The Negation of a Conditional Statement ~(IF my car is in the repair shop, THEN I cannot get the class) My car is in the repair shop and I can get to class ~(IF Sara lives in Athens, THEN she lives in Greece) Sara lives in Athens and she does not live in Greece http://arief.ismy.web.id

The Contrapositive of a Conditional Statements Definition The contrapositive of a conditional statement of the form “IF p THEN q” is “IF ~q THEN ~p” The contrapositive of p  q is ~q  ~p A conditional statement is logically equivalent to its contrapositive. Construct the truth table http://arief.ismy.web.id

Example IF Howard can swim across the lake, THEN Howard can swim to the island IF Howard cannot swim to the island, then Howard cannot swim across the lake IF today is Easter, THEN tomorrow is Monday IF tomorrow is not Monday, THEN today is not Easter http://arief.ismy.web.id

The Converse and Inverse of a Conditional Statement Definition Suppose a conditional statement of the form “IF p THEN q” is given. The converse is “IF q THEN p” The inverse is “IF ~p THEN ~q” Symbolically The converse of p  q is q  p The inverse of p  q is ~p  ~q Are they logically equivalent?? http://arief.ismy.web.id

pq~p~q p  qq  p~p  ~q TTFFTTT TFFTFTT FTTFTFF FFTTTTT http://arief.ismy.web.id

Example: Converse, Inverse IF Howard can swim across the lake, THEN Howard can swim to the island Converse: IF Howard can swim to the island THEN Howard can swim across the lake Inverse: IF Howard cannot swim across the lake, THEN Howard cannot swim to the island. IF today is Easter, THEN tomorrow is Monday Converse: IF tomorrow is Monday, THEN today is Easter Inverse: IF today is not Easter, THEN tomorrow is not Monday http://arief.ismy.web.id

Only If Definition p and q are statements, p only if q means “IF not q THEN not p” Or, equivalently, “IF p THEN q” John will break the world’s record for the mile run ONLY IF he runs the mile in under four minutes. IF John does not run the mile in under four minutes, THEN he will not break the world’s record IF John breaks the world’s record, THEN he will have run the mile in under four minutes http://arief.ismy.web.id

Biconditional Definition Given statement variables p and q, the biconditional of p and q is “ p if, and only if, q” and is denoted p  q. The words if and only if are sometimes abbreviated iff. http://arief.ismy.web.id

Biconditional Is “ p if, and only if, q” logically equivalent with “ p if q “ and “ p only if q” ? p  q  (q  p)  (p  q) Construct the truth table http://arief.ismy.web.id

p  q  (q  p)  (p  q) http://arief.ismy.web.id

Necessary and Sufficient Conditions Definition If r and s are statements: –r is a sufficient condition for s means “if r then s” –r is a necessary condition for s means “if not r then not s” or “if s then r” r is a necessary and sufficient condition for s means “r if, and only if, s”. http://arief.ismy.web.id