In each part, a list of hypotheses is given. These hypotheses are assumed to be true....

  

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In each part, a list of hypotheses is given. These hypotheses are assumed to be true. Using tautologies from Example 3.12, you are to establish tbe desired conclusion. Indicate which tautology you are using to justify each step. (a) Hypotheses: r ~s, t s Conclusion: r t (b) Hypotheses: r, t, (rA s)t Conclusion: s (c) Hypotbeses: r s, r t, ~t> u, vS Conclusion: ~D v u 3.12 EXAMPLES The following tautologies are useful in constructing proofs. The first two indicate, for example, that an "if and only if" theorem pq can be proved by establishing pq and its converse qp or by showing pq and its inverse -p -q. The lettere is used to represent a statement that is always false. Such a statement is called a contradiction. While this list of tautologies need not be memorized, it will be helpful if each tautology is studied carefully to see just what it is saying (a) (pq)es [(p q) a (q p)] (b) (pq) e [(p=) A (~p -q)] (c) (pq) (~q-p) (d) pV P (e) (pA ~p)c (f) (~pc)p (g) [(p A ng) c] (p q) (h) [p A (p=q)14 (1) [~qA (p> q)]= -p ) [~p A (p v q))=4 (k) (pA q) P (1) [=4) A (yr)]= (pr) (m) [P, P2) (P2 Pa) A (n) [p a q) r] [p(4 r)] (0) [(p4) A (r s) A (p v r)] (q v s) (p) [p>(g v r)][(p A - 4) r] (q) [(pr) A (q r)][p v g r] .. A (P.-P] (P, p.) In each part, a list of hypotheses is given. These hypotheses are assumed to be true. Using tautologies from Example 3.12, you are to establish tbe desired conclusion. Indicate which tautology you are using to justify each step. (a) Hypotheses: r ~s, t s Conclusion: r t (b) Hypotheses: r, t, (rA s)t Conclusion: s (c) Hypotbeses: r s, r t, ~t> u, vS Conclusion: ~D v u 3.12 EXAMPLES The following tautologies are useful in constructing proofs. The first two indicate, for example, that an "if and only if" theorem pq can be proved by establishing pq and its converse qp or by showing pq and its inverse -p -q. The lettere is used to represent a statement that is always false. Such a statement is called a contradiction. While this list of tautologies need not be memorized, it will be helpful if each tautology is studied carefully to see just what it is saying (a) (pq)es [(p q) a (q p)] (b) (pq) e [(p=) A (~p -q)] (c) (pq) (~q-p) (d) pV P (e) (pA ~p)c (f) (~pc)p (g) [(p A ng) c] (p q) (h) [p A (p=q)14 (1) [~qA (p> q)]= -p ) [~p A (p v q))=4 (k) (pA q) P (1) [=4) A (yr)]= (pr) (m) [P, P2) (P2 Pa) A (n) [p a q) r] [p(4 r)] (0) [(p4) A (r s) A (p v r)] (q v s) (p) [p>(g v r)][(p A - 4) r] (q) [(pr) A (q r)][p v g r] .. A (P.-P] (P, p.)

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Ans Y Ys Con lusim Criven NY is drue Y s false NYAS Cant imply a fals... View the full answer