Logic Lecture Notes PDF

Lecture 5-8  Using law of logic, simplify the statement form p ∨ [~(~p ∧ q)] Solution:  p ∨ [~(~p ∧ q)] ≡ p ∨ [~(~p) ∨ (~q)] DeMorgan’s Law  ≡ p ∨ [p∨(~q)] Double Negative Law: ~(~p) ≡ p  ≡ [p ∨ p]∨(~q) Associative Law for ∨  ≡ p ∨ (~q)...

Lecture 5-8  Using law of logic, simplify the statement form p ∨ [~(~p ∧ q)] Solution:  p ∨ [~(~p ∧ q)] ≡ p ∨ [~(~p) ∨ (~q)] DeMorgan’s Law  ≡ p ∨ [p∨(~q)] Double Negative Law: ~(~p) ≡ p  ≡ [p ∨ p]∨(~q) Associative Law for ∨  ≡ p ∨ (~q) Idempotent Law: p ∨ p ≡ p That is the simplified statement form.  Example: Using Laws of Logic, verify the logical equivalence ~ (~ p ∧ q) ∧ (p ∨ q) ≡ p Solution:  ~(~p ∧ q) ∧ (p∨q) ≡ (~(~p) ∨ ~q) ∧(p ∨ q) DeMorgan’s Law  ≡ (p ∨ ~q) ∧ (p∨q) Double Negative Law  ≡ p ∨ (~q ∧ q) Distributive Law  ≡p∨c Negation Law  ≡p Identity Law Introduction Consider the statement:  "If you earn an A in Math, then I'll buy you a computer." This statement is made up of two simpler statements: p: "You earn an A in Math" q: "I will buy you a computer." The original statement is then saying :  if p is true, then q is true, or, more simply, if p, then q.  We can also phrase this as p implies q. It is denoted by p → q.  If p and q are statement variables, the conditional of q by p is “If p then q”or “p implies q” and is denoted p → q.  p → q is false when p is true and q is false; otherwise it is true.  The arrow "→ " is the conditional operator.  In p → q, the statement p is called the hypothesis (or antecedent) and q is called the conclusion (or consequent). Let p and q be propositions:  p = “you get an A on the final exam”  q = “you do every exercise in this book”  r = “you get an A in this class”  1.To get an A in this class it is necessary for you to get an A on the final. SOLUTION p → r  2.You do every exercise in this book; You get an A on the final, implies, you get an A in the class. SOLUTION p ∧ q → r  3. Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class.  SOLUTION p ∧ q → r Let p, q, and r be the propositions:  p = “you have the flu”  q = “you miss the final exam”  r = “you pass the course” Express the following propositions as an English sentence.  1. p → q If you have flu, then you will miss the final exam.  2. ~q → r If you don’t miss the final exam, you will pass the course.  3. ~p ∧ ~q→ r If you neither have flu nor miss the final exam, then you will pass the course. HIERARCHY OF OPERATIONS  FOR LOGICAL CONNECTIVES  ~ (negation)  ∧ (conjunction), ∨ (disjunction)  → (conditional) Example: Construct a truth table for the statement form (p →q)∧(~ p → r) LOGICAL EQUIVALENCE INVOLVING IMPLICATION use truth table to show p→q ≡ ~q → ~p  Since p→q ≡ ~ p∨q  So ~ (p → q) ≡ ~ (~ p ∨ q)  ≡ ~ (~ p) ∧ (~ q) by De Morgan’s law  ≡p∧~q by the Double Negative law  Thus the negation of “if p then q” is logically equivalent to “p and not q”.  Accordingly, the negation of an if-then statement does not start with the word if. EXAMPLES  Write negations of each of the following statements:  1.If Ali lives in Pakistan then he lives in Lahore.  2.If my car is in the repair shop, then I cannot get to class.  3.If x is prime then x is odd or x is 2.  4.If n is divisible by 6, then n is divisible by 2 and n is divisible by 3.  SOLUTIONS:  1. Ali lives in Pakistan and he does not live in Lahore.  2. My car is in the repair shop and I can get to class.  3. x is prime but x is not odd and x is not 2.  4. n is divisible by 6 but n is not divisible by 2 or by 3.  The inverse of the conditional statement p → q is ~p → ~q, A conditional and its inverse are not equivalent as could be seen from the truth table.  1. If today is Friday, then 2 + 3 = 5. If today is not Friday, then 2 + 3 ≠ 5.  2. If it snows today, I will ski tomorrow. If it does not snow today I will not ski tomorrow.  3. If P is a square, then P is a rectangle. If P is not a square then P is not a rectangle.  4. If my car is in the repair shop, then I cannot get to class. If my car is not in the repair shop, then I shall get to the class.  The converse of the conditional statement p → q is q →p.  A conditional and its converse are not equivalent. i.e., → is not a commutative operator.  1.If today is Friday, then 2 + 3 = 5. If 2 + 3 = 5, then today is Friday.  2.If it snows today, I will ski tomorrow. I will ski tomorrow only if it snows today.  3. If P is a square, then P is a rectangle. If P is a rectangle then P is a square.  4. If my car is in the repair shop, then I cannot get to class.  If I cannot get to the class, then my car is in the repair shop.  The contra-positive of the conditional statement p → q is ~ q → ~ p, A conditional and its contra-positive are equivalent. Symbolically p→q ≡ ~q → ~p  1.If today is Friday, then 2 + 3 = 5. If 2 + 3 ≠ 5, then today is not Friday.  2.If it snows today, I will ski tomorrow. I will not ski tomorrow only if it does not snow today.  3. If P is a square, then P is a rectangle. If P is not a rectangle then P is not a square.  4. If my car is in the repair shop, then I cannot get to class. If I can get to the class, then my car is not in the repair shop.  If p and q are statement variables, the biconditional of p and q is “p if and only if q”.  It is denoted p↔q. “if and only if” is abbreviated as iff.  The double headed arrow " ↔" is the biconditional operator. Identify which of the following are True or false?  1.“1+1 = 3 if and only if earth is flat” TRUE  2. “Sky is blue iff 1 = 0” FALSE  3. “Milk is white iff birds lay eggs” TRUE  4. “33 is divisible by 4 if and only if horse has four legs” FALSE  5. “x > 5 iff x2 > 25” FALSE  p↔q is also expressed as:  o “p is necessary and sufficient for q”  o “If p then q, and conversely”  o “p is equivalent to q” Rephrase the following propositions in the form “p if and only if q” in English.  1. If it is hot outside, you buy an ice cream cone, and if you buy an ice cream cone, it is hot outside. Sol You buy an ice cream cone if and only if it is hot outside.  2. For you to win the contest it is necessary and sufficient that you have the only winning ticket. Sol You win the contest if and only if you hold the only winning ticket.  3. If you read the news paper every day, you will be informed and conversely. Sol You will be informed if and only if you read the news paper every day.  4. It rains if it is a weekend day, and it is a weekend day if it rains. Sol It rains if and only if it is a weekend day.  5. The train runs late on exactly those days when I take it. Sol The train runs late if and only if it is a day I take the train.  6. This number is divisible by 6 precisely when it is divisible by both 2 and 3. Sol This number is divisible by 6 if and only if it is divisible by both 2 and 3. Example: Show that ~p ↔ q and p ↔ ~q are logically equivalent.  Show that ~(p⊕q) and p↔q are logically equivalent.  1.Commutative Law: p↔q≡q↔p  2.Implication Laws: p → q ≡ ~ p ∨ q ≡ ~ ( p ∧ ~ q)  3.Exportation Law: (p ∧ q)→r ≡ p →(q →r)  4.Equivalence: p ↔ q ≡ (p →q)∧(q →p)  5.Reductio ad absurdum p →q ≡ (p ∧ ~q) →c Example: Rewrite the statement forms without using the symbols → or ↔  1. p ∧ ~ q→ r  2. ( p→ r ) ↔ ( q → r ) 1. p∧~q→r ≡ (p ∧ ~q)→ r Order of operations ≡ ~ (p ∧ ~ q) ∨ r Implication law 2. (p→r)↔(q →r) ≡ (~p ∨ r) ↔ (~q ∨ r) Implication law ≡ [(~p ∨ r) →(~q ∨ r)] ∧ [(~q ∨ r) →(~p ∨ r)] Equivalence of biconditional ≡ [~(~p ∨ r) ∨ (~q ∨ r)] ∧ [~(~q ∨ r) ∨ (~p ∨ r)] Implication law  Example: Rewrite the statement form ~p ∨ q → r ∨ ~q to a logically equivalent form that uses only ~ and ∧.  Solution: STATEMENT REASON  ~p ∨ q → r ∨ ~q Given statement form  ≡ (~p ∨ q) → (r ∨ ~q) Order of operations  ≡ ~[(~p ∨ q) ∧ ~ (r ∨ ~q)] Implication law  p→q ≡ ~(p∧~ q)  ≡ ~[~(p ∧ ~q) ∧ (~r ∧ q)] De Morgan’s law  Example: Show that ~(p→q) → p is a tautology without using truth tables. STATEMENT REASON  ~(p→q) → p Given statement form  ≡ ~[~(p ∧ ~q)] → p Implication law p→q ≡ ~(p ∧ ~q)  ≡ (p ∧ ~q) → p Double negation law  ≡ ~(p ∧ ~q) ∨ p Implication law p→q ≡ ~p ∨ q  ≡ (~p ∨ q) ∨ p De Morgan’s law  ≡ (q ∨ ~p) ∨ p Commutative law of ∨  ≡ q ∨ (~p ∨ p) Associative law of ∨  ≡q∨t Negation law  ≡t Universal bound law Suppose that p and q are statements so that p→q is false. Find the truth values  of each of the following:  1.~ p → q  2.p ∨ q  3.q ↔ p Hint: ( p→q is false when p is true and q is false.)  1.TRUE  2.TRUE  3.FALSE