Compound Propositions
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1 Discrete Structures
2 Compound Propositions Producing new propositions from existing propositions. Logical Operators or Connectives 1. Not 2. And 3. Or 4. Exclusive or 5. Implication 6. Biconditional
3 Truth Table (NOT) Unary Operator, Symbol: p true false p false true
4 Conjunction (AND) Definition Let p and q be propositions. The conjunction of p and q, denoted by p q, is the proposition p and q. The conjunction p q is true when p and q are both true and is false otherwise.
5 Examples 1. Find the conjunction of the propositions p and q, where The conjunction is p : Today is Friday. q : It is raining today. p q : Today is Friday and it is raining today.
6 Truth Table (AND) Binary Operator, Symbol: p q p q true true true true false false false true false false false false
7 Disjunction (OR) Definition Let p and q be propositions. The disjunction of p and q, denoted by p q, is the proposition p or q. The disjunction p q is false when both p and q are false and is true otherwise.
8 Examples 1. Find the disjunction of the propositions p and q, where The disjunction is p : Today is Friday. q : It is raining today. p q : Today is Friday or it is raining today.
9 Truth Table (OR) Binary Operator, Symbol: p q p q true true true true false true false true true false false false
10 Exclusive OR (XOR) Definition Let p and q be propositions. The exclusive or of p and q, denoted by p q, is the proposition p q. The exclusive or, p q, is true when exactly one of p and q is true and is false otherwise.
11 Examples 1. Find the exclusive or of the propositions p and q, where p : Atif will pass the course CSC102. q : Atif will fail the course CSC102. The exclusive or is p q : Atif will pass or fail the course CSC102.
12 Truth Table (XOR) Binary Operator, Symbol: p q p q true true false true false true false true true false false false
13 Examples (OR vs XOR) The following proposition uses the (English) connective or. Determine from the context whether or is intended to be used in the inclusive or exclusive sense. 1. Nabeel has one or two brothers. A person cannot have both one and two brothers. Therefore, or is used in the exclusive sense.
14 Examples (OR vs XOR) 2. To register for BSC you must have passed the qualifying exam or be listed as an Math major. Presumably, if you have passed the qualifying exam and are also listed as an Math major, you can still register for BCS. Therefore, or is inclusive.
15 Composite Statements Statements and operators can be combined in any way to form new statements. p q p q ( p) ( q) true true false false false true false false true true false true true false true false false true true true
16 Tautologies and Contradictions Tautology is a statement that is always true regardless of the truth values of the individual logical variables Examples: R ( R) (P Q) ( P) ( Q) If S T is a tautology, we write S T. If S T is a tautology, we write S T.
17 Tautologies and Contradictions A Contradiction is a statement that is always false regardless of the truth values of the individual logical variables Examples R ( R) ( (P Q) ( P) ( Q)) The negation of any tautology is a contradiction, and the negation of any contradiction is a tautology.
18 Logical Equivalence Commutative laws: p q = q p, p q = q p Associative laws: (p q) r = p (q r), (p q) r = p (q r) Distributive laws: p (q r) = (p q) (p r) p (q r) = (p q) (p r) Identity laws: p t = p, p c = p Negation laws: p ~p = t, p ~p = c Double negative law: ~(~p) = p Idempotent laws: p p = p, p p = p De Morgan s laws: ~(p q) = ~p ~q, ~(p q) = ~p ~q Universal bound laws: p t = t, p c = c Absorption laws: p (p q) = p, p (p q) = p Negation of t and c: ~t = c, ~c = t
19 Exercises Simplify: ~(~p q) (p q) Simplify : (p (~p q)) ~(q ~r)
20 Conditional Statements Contrapositive p q is another conditional statement ~q ~p A conditional statement is equivalent to its contrapositive The converse of p q is q p The inverse of p q is ~p ~q Conditional statement and its converse are not equivalent Conditional statement and its inverse are not equivalent
21 Conditional Statements The converse and the inverse of a conditional statement are equivalent to each other p only if q means ~q ~p, or p q
22 Biconditional of p and q means p if and only if q and is denoted as p q 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
23 Conditional Statements Implication Definition: Let p and q be propositions. The conditional statement p q, is the proposition If p, then q. The conditional statement p q is false when p is true and q is false and is true otherwise. where p is called hypothesis, antecedent or premise. q is called conclusion or consequence
24 Implication (if - then) Binary Operator, Symbol: P Q P Q true true true true false false false true true false false true
25 Conditional Statements Biconditional Statements Definition: Let p and q be propositions. The biconditional statement p q, is the proposition p if and only if q. The biconditional (bi-implication) statement p q is true when p and q have same truth values and is false otherwise.
26 Biconditional (if and only if) Binary Operator, Symbol: P Q P Q true true true true false false false true false false false true
27 Equivalent Statements P Q (P Q) ( P) ( Q) (P Q) ( P) ( Q) true true false false true true false true true true false true true true true false false true true true Two statements are called logically equivalent if and only if (iff) they have identical truth tables The statements (P Q) and ( P) ( Q) are logically equivalent, because (P Q) ( P) ( Q) is always true.
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