LOGIC AND TRUTH TABLES



Name ________________________Worksheet 1.2Logic and Truth TablesWe can represent the truth of expressions in a tabular form called “truth tables.” These tables consider all cases and can add great insight into otherwise complicated expressions. Here are examples of some of most basic truth tables.The truth table for negation (“not”)p? pTFFTThe truth table for disjunction (“or”)pqp qTTTTFTFTTFFF1. Complete the following truth tables(a) Truth table for conjunction (“and”)pqp qTTTFFTFF(b) Truth table for biconditional (“if and only if”)pqp ? qTTTFFTFFWhy do we say that the biconditional defines “logical equivalence”?2. The truth table for conditional (“if...then...”) can be quite confusing. Here it is.pqp → qTTTTFFFTTFFTConsider the example “For every integer n, if n > 2 then n2 > 4.” In this case p represents “n > 2” and q represents “n2 > 4.” Thus if n = 3 then p is true and q is true.(a) Find another value of n that makes p true and q true.(b) Find a value for n that makes p false and q false.(c) Find a value for n that makes p false and q true.(d) Find a value for n that make p true and q false. Explain any problems you have. Why are you having these problems?The third row, corresponding to part (c), frustrates many people—you are not alone. The truth has a special name. It is called “vacuously true.”3. Complete the following truth tables(a) pq(? p) ? qTTTFFTFF(b) pq(? p) qTTTFFTFF(c) pqrq → rp → (q → r)TTTTTFTFTTFFFTTFTFFFTFFF4. Draw up and complete the truth tables for the following expressions. Please follow all of the conventions that you have observed above (such as listing p before q, etc.)(a) p (? p)(b) (? p) (? q)5. An expression is a tautology if it is always true, regardless the truth of its component statements. Thus, the last column in its truth table will be all T’s. A statement is a contradiction if the last column is always F—it is never true. (a) Show that (p q) → p is a tautology by completing this truth table.pqp q(p q) → pTTTTTFTFFTFTFFFF(b) Show that p (? p) is a contradiction.pp (? p)TF6. Logically equivalent expressions have the same truth tables. Show that the following pairs of statements are logically equivalent. (a) (? p) q and (p q) (You saw this on the previous worksheet. Where?)(b) p → q and (? q) → (? p) (The second statement is called the Contrapositive)(c) (p q) r and (p r) (q r) (Distribution of sorts)(d) ? (p q) and (? p) (? q) (This, and part (e) is called De Morgan’s Law)(e) ? (p q) and (? p) (? q) ................
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