Goal: 1. Review propositional logic, focus on: conditional, - - PDF document

Lesson #2

• CISC2100
• Fall 2017
• Goal:
• 1.

Review propositional logic, focus on: conditional, tautology/contradiction, logical equivalence (contrapositive, converse & inverse) 2. New terminology and historic background: English => symbols (statement form, or propositional form 3. You should be able to: 1. Convert between English sentences and statement forms (in particular, if, only if, if and

• nly if, sufficient condition, necessary condition)

2. Draw truth table for compound statement forms 3. Determine whether a statement form is tautology/contradiction 4. Determine whether two statement forms are logically equivalent or not 5. Simplify statement forms using logical equivalence (Theorem 2.1.1), including simplify the negation of a statement form

• History:
• Aristotle (384 B.C.-322 B.C.) => Lebinez (17th century) => George Boole, Augustus De

Morgan, Russel’s paradox

• Mathematical/symbolic logic

English statement

• letters

and, or, not, if … then … ^, v, ~, =>, <=>

• Application to Computer Science: Digital Logic, AI
• Group Exercises
• 1. Review of logic connectives (operations):
• Operation Symbol Example compound statements Truth table C++ operator

negation

• Conjunction (and)
• Disjunction (or)
• Exclusive or
• Conditional (if…then…)

Biconditional

• Order of operations (precedence level)

~ first, ^ next, V next —>, <—> next

• 2. English statement to statement form
• A statement form (or propositional form) is an expression made up of statement variables

(such as p, q, and r) and logical connectives (such as above) that becomes a statement when actual statements are substituted for the component variables.

• For example, p => q, if p is substituted with “you show up on Monday for work”, q with “you get

the job”, p=>q becomes a statement “if you show up on Monday for work, then you get the job.

• if p is substituted with “it rains”, q with “the ground is wet”, then ….
• We can say the above two statements have same form.
• Exercise: What’s the statement forms of the following statement: “John is not 6 feet tall or he

weight less than 200 pounds”.

• “1<x<=40” where x stands for a particular variable whose value is 3
• 3. Review: Tautology and contradiction, logical equivalence
• Common tool: write truth-tables for the statement form

a) identify variables, one column per variable b) one column per

• Example. ~(p^q)= ~p ^ ~q
• Exercise: p->q = ~q V p

• Exercise: distributive law
• Exercise: p->q = ~q -> ~p
• p->q =? ~p -> ~q
• 4. Contrapositive, converse, inverse of a conditional statement
• Statement: if my car is in the repair shop, then I cannot get to class.
• p -> q
• 1) contrapositive:
• 2) converse
• 3) inverse
• 5. Summary of all logical equivalence
• 6. Usage of the logical equivalence:
• simplify a statement form, negate a statement form