Exercise 2. Is x2+5x+6 a prime number, where x is a positive integer? Exercise 3. Build the truth table of (pq)(qp), where p and q are two propositional formulae? Is the previous formula a tautology? Exercise 4 (Ex. 11, Chapter 1 of [Martin; 2011]). In each case below, say whether the given statement is true for the universe (0,1)={xR0y)) b. x(y(xy)) c. y(x(x>y)) d. y(x(xy)) Exercise 5 (Ex. 12, Chapter 1 of [Martin; 2011]). (a) How many elements are there in the set {,{},{,{}},{,{{,{},{,{}}}}}} ? (b) Describe precisely the algorithm you used to answer part (a). Exercise 6 (Ex. 17, Chapter 1 of [Martin; 2011]). In each case below, say whether the indicated function is one-to-one and what its range is. a. m:NN defined by m(x)=min(x,2) b. M:NN defined by M(x)=max(x,2) c. s:NN defined by s(x)=m(x)+M(x) d. f:N{0}2N, where f(n) is the set of prime factors of n e. (Here A is the set of all finite sets of primes and B is the set N{0}.) Let g:AB, where g(S) is the product of the elements of S. (The product of the elements of the empty set is 1.) Exercise 7 (Ex. 23, Chapter 1 of [Martin; 2011]). In each case below, a relation on the set {1,2,3} is given. Of the three properties, reflexivity, symmetry, and transitivity, determine which ones the relation has. Give reasons. (a) R={(1,3),(3,1),(2,2)} (b) R={(1,1),(2,2),(3,3),(1,2)} (c) R= Exercise 8 (Ex. 40, Chapter 1 of [Martin; 2011]). Suppose A is a set having n elements. a. How many relations are there on A ? b. How many reflexive relations are there on A ? c. How many symmetric relations are there on A ? d. How many relations are there on A that are both reflexive and symmetric