13. Definition 1.34 in the textbook defines the term semantic entailment which I will explain here. Suppose we have a sequent 01, 02, ..., O., V, e.g., in (P V - Vr), (-2 VqV -r) - (r^-9), 0. = (p V - Vr), Q2 =( p V 9V-r), and V = (r 1-9). An interpretation for the formulas 01, 02, ..., 0, is created by assigning t or f to each of the atoms of the formulas. For example, let I be the interpretation I = {p=t, q=f, r=f}. (a) How many interpretations are there for a formula I which consists of seven unique atoms? (b) How many interpretations are there for the example formulas 04 and 0,? (c) How many interpretations are there for the formulas of the sequent in Q3(a)? A valuation occurs when given the assignment of I to the atoms of each 01, 02, On, we perform each logic operation of 01, 02, On to determine if Q: evaluates to t or f. For example, V[0] = (p V - V r) = (t V -f V f) = t and V[0] = (-p va V -r) = (-t VfV -f) = (f VfV t) = t. = (c) For the example formula On, what is V [:] when I = {p = t, q = f, r = f}? (d) For the example formula 02, what is V:[02] given the same I as in (c). (e) Let I = {v = t, u = t, t = f} be an interpretation for the formula of Q3(a). What are VI[01], VI[02], and VI[V]? Now we can come back to the semantic entailment definition. It states: If for all valuations in which all of 01, 02, ..., O, evaluate to t, V evaluates to t as well, then we say 01, 02, ..., O., I holds and we call = the semantic entailment relation. For example, suppose we have the sequent (a V-6), (a 4 b) 6. The question is does (a V -b), (a ^ b) = b. First, let 01 = (a V -b), 02 = (a A b), and V = b. We need to find all valuations in which 0, and 0. evaluate to t. To do that we need to determine how many different interpretations there are for 01 and 02. Since both of Q, and 02 contain 2 unique atoms, there will be 4 interpretations. Let them be I1 = {a = f, b=f}, 12 = {a = f, b = t}, 13 = {a=t, b = f}, and 14 = {a=t, b = t}. (f) Write all of the interpretations for the formulas of the sequent in Q3(a) and label them I1, I2, ..., Im. Next, we need to evaluate Oand 0, for each interpretation, i.e., we need to compute V [01], V:[0,], Va[O:], Vr[], VA[0,], VA[02], V,[O], VA[], as well as the value of V for each interpretation. This can be accomplished by draw- ing the truth table for 01, 02, and V: a b -6 (a V -b)(a Ab 1 1 V.[0:1 V.[:] f f t t f t f f f t f f f f t 2 3 f f t f t t f t f 4 t t f t t t t t There is only one valuation in which V [01] = t, V:[0] = t and y = t and that is the one for 14. Because in this valuation, all of 0, and 0, evaluated to t and I evaluated to t as well, then we can say 01, 0, - V holds. (g) Draw a truth table using a similar format to the above truth table for sequent of Q3(a). (h) Does v (u t), v u, E Vt hold? 13. Definition 1.34 in the textbook defines the term semantic entailment which I will explain here. Suppose we have a sequent 01, 02, ..., O., V, e.g., in (P V - Vr), (-2 VqV -r) - (r^-9), 0. = (p V - Vr), Q2 =( p V 9V-r), and V = (r 1-9). An interpretation for the formulas 01, 02, ..., 0, is created by assigning t or f to each of the atoms of the formulas. For example, let I be the interpretation I = {p=t, q=f, r=f}. (a) How many interpretations are there for a formula I which consists of seven unique atoms? (b) How many interpretations are there for the example formulas 04 and 0,? (c) How many interpretations are there for the formulas of the sequent in Q3(a)? A valuation occurs when given the assignment of I to the atoms of each 01, 02, On, we perform each logic operation of 01, 02, On to determine if Q: evaluates to t or f. For example, V[0] = (p V - V r) = (t V -f V f) = t and V[0] = (-p va V -r) = (-t VfV -f) = (f VfV t) = t. = (c) For the example formula On, what is V [:] when I = {p = t, q = f, r = f}? (d) For the example formula 02, what is V:[02] given the same I as in (c). (e) Let I = {v = t, u = t, t = f} be an interpretation for the formula of Q3(a). What are VI[01], VI[02], and VI[V]? Now we can come back to the semantic entailment definition. It states: If for all valuations in which all of 01, 02, ..., O, evaluate to t, V evaluates to t as well, then we say 01, 02, ..., O., I holds and we call = the semantic entailment relation. For example, suppose we have the sequent (a V-6), (a 4 b) 6. The question is does (a V -b), (a ^ b) = b. First, let 01 = (a V -b), 02 = (a A b), and V = b. We need to find all valuations in which 0, and 0. evaluate to t. To do that we need to determine how many different interpretations there are for 01 and 02. Since both of Q, and 02 contain 2 unique atoms, there will be 4 interpretations. Let them be I1 = {a = f, b=f}, 12 = {a = f, b = t}, 13 = {a=t, b = f}, and 14 = {a=t, b = t}. (f) Write all of the interpretations for the formulas of the sequent in Q3(a) and label them I1, I2, ..., Im. Next, we need to evaluate Oand 0, for each interpretation, i.e., we need to compute V [01], V:[0,], Va[O:], Vr[], VA[0,], VA[02], V,[O], VA[], as well as the value of V for each interpretation. This can be accomplished by draw- ing the truth table for 01, 02, and V: a b -6 (a V -b)(a Ab 1 1 V.[0:1 V.[:] f f t t f t f f f t f f f f t 2 3 f f t f t t f t f 4 t t f t t t t t There is only one valuation in which V [01] = t, V:[0] = t and y = t and that is the one for 14. Because in this valuation, all of 0, and 0, evaluated to t and I evaluated to t as well, then we can say 01, 0, - V holds. (g) Draw a truth table using a similar format to the above truth table for sequent of Q3(a). (h) Does v (u t), v u, E Vt hold