Foundations of Mathematics Worksheet 2

Size: px

Start display at page:

Download "Foundations of Mathematics Worksheet 2"

Baldwin Wilkins
6 years ago
Views:

1 Foundations of Mathematics Worksheet 2 L. Pedro Poitevin June 24, What are the atomic truth assignments on {a 1,..., a n } that satisfy: (a) The proposition p = ((a 1 a 2 ) (a 2 a 3 ) (a n 1 a n ))? (b) The proposition q = (p (a n a 1 ))? (c) The proposition r = 1 i n 1 j n i j (a i a j )? Write down disjunctive normal forms for p, q, and r. 2. Consider the set of propositional variables P = {a 1,..., a n }. (a) Show that the following proposition is a tautology: ( ) (a i a j ) ( ) a j. 1 i<j n (b) Which atomic truth assignments on P make the following proposition false: ( ) a i ( ) a j? (c) Show that the preceding proposition is logically equivalent to ( a i ) a j. j i 3. ( ) A safe has n locks and can be opened only when all n of the locks are open. Five people, a, b, c, d, and e are to receive keys to some of the locks. Each key can be duplicated any number of times. Find the smallest value of n, and a corresponding distribution of keys to the five people, so that the safe can be opened if and only if at least one of the following situations applies: (a) a and b are present together; j i j i

2 (b) a, c and d are present together; (c) b, d, and e are present together. 4. The language L consists of a unary function symbol f and a binary function symbol g. Consider the following sentences: ϕ 6 : x y(f(g(x, y)) = f(x)) x y(f(g(x, y)) = f(x)) y x(f(g(x, y)) = f(x)) x y(f(g(x, y)) = f(x)) x y(f(g(x, y)) = f(x)) y x(f(g(x, y)) = f(x)). Consider the four structures whose underlying set is Z +, where g is interpreted by the map (m, n) m + n, and where f is interpreted respectively by (a) the constant map equal to 103; (b) the map n n mod 4; (c) the map n min(n 2 + 1, 19); (d) the map n { 1, if n = 1 the smallest prime divisor of n, if n > 1. For each of the six sentences, determine whether it is satisfied or not in each of the four structures. 5. The language L consists of a unary predicate symbol P and a binary predicate symbol R. Consider the following six sentences: ϕ 6 : x y z((p (x) R(x, y)) P (y) R(y, z)) x z((r(z, x) R(x, z)) yr(x, y)) y( z tr(t, z) x(r(x, y) R(x, y))) x y((p (y) R(y, x)) ( u(p (u) R(y, x)) R(x, y))) x y((p (x) R(x, y)) ((P (y) R(y, x)) z( R(z, x) R(y, z)))) z u x y((r(x, y) P (u)) (P (y) R(z, x))). For each of these sentences, determine whether or not it is satisfied in each of the three L-structures defined below: (a) The underlying set is N, the interpretation of R is the usual order relation, the interpretation of P is the set of even integers. (b) The underlying set is P(N) (the set of all subsets of N), the interpretation of R is the inclusion relation, and the interpretation of P is the collection of finite subsets of N. (c) The underlying set is R, the interpretation of R is the set of pairs (a, b) R 2 such that b = a 2, the interpretation of P is the subset of rational numbers. 2

3 6. The language L has two unary function symbols f and g. (a) Find three sentences σ 1, σ 2, σ 3 of L such that for every L-structure A = (A, f, g) we have A σ 1 if and only if f = g and f is a constant map; A σ 2 if and only if f(a) g(a); A σ 3 if and only if f(a) g(a) contains a single element. (b) Consider the following five L-sentences: xf(x) = g(x) x yf(x) = g(y) x yf(x) = g(y) x yf(x) = g(y) x yf(x) = g(y). Find a model for each of the following six sentences: a ϕ 1 ϕ 2 ; b ϕ 2 ; c ϕ 1 ϕ 3 ; d ϕ 1 ϕ 4 ; e ϕ 3 ϕ 4 ϕ 5 ; f ϕ Let L be a language and let A[x, y] be an arbitrary L-formula with two free variables. (a) Is the sentence x ya[x, y] y xa[x, y] satisfied in any L-structure? (b) Is the sentence y xa[x, y] x ya[x, y] satisfied in any L-structure? 8. Show that the following sets are 0-definable in the corresponding structures: (a) The ordering relation {(m, n) N 2 m < n} in (N, 0, +). (b) The set of prime numbers in the semiring N = (N, 0, 1, +, ). (c) The set {2 n n N} in the semiring N. (d) The set {a R f is continuous at a} in (R, <, f) where f : R R is any function. (e) The set {m Z : m 0} in the ring (Z, 0, 1,, +, ) 9. In all the of the languages considered in this exercise, R is a binary relation symbol, and are binary function symbols, and c and d are constant symbols. We will write x y and x y rather than xy and xy, respectively. We will also abbreviate x x by x 2. a In each of the following six cases (1 i 6), a language L i and two L i -structures A i and B i are given to you, and you are asked to find an L i -sentence that is true in A i and false in B i. 3

4 (a) L 1 = {R}, A 1 = (N, ), B 1 = (Z, ); (b) L 2 = {R}, A 2 = ((Q), ), B 2 = (Z, ); (c) L 3 = { }, A 3 = ((N), ), B 3 = (P(N), ); (d) L 4 = {c, }, A 4 = (N, 1, ), B 4 = (Z, 1, ); (e) L 5 = {c, d,, }, A 5 = ((R), 0, 1, +, ), B 5 = (Q, 0, 1, +, ); (f) L 6 = {R}, A 6 = ((Z), 2 ), B 6 = (Z, 3 ); (where and + denote the usual operations of multiplication and addition, denotes the operation of intersection, and p is the relation of congruence modulo p.) b For each of the following sentences of the language {c,,, R}, find a model of the sentence as well as a model of its negation. u v x( v = c u (v x) = c) u v w x( w = c u (v x) (w x 2 ) = c) x y z(r(x, x) ((R(x, y) R(y, z)) R(x, z)) (R(x, y) R(y, x))) x y z(r(x, y) R(x z, y z)) x y(r(x, y) R(y, x)). 10. The language L consists of a single binary relation symbol R. Consider the L- structure A whose underlying set is A = {n N n 2} and in which R is interpreted by the relation divides, i.e. R A is defined for all integers m, n 2 by the condition: (m, n) R A if and only if m divides n. a For each of the following L-formulas (with one free variable x), describe the set of elements of A that satisfy it. y(r(y, x) x = y) y z((r(y, x) R(z, x) (R(y, z) R(z, y))) y z(r(y, x) (R(z, y) R(x, z))) t y z(r(t, x) (R(y, t) R(z, y) R(t, z))). b Write an L-formula ψ[x, y, z, t] such that for all a, b, c, and d in A, the structure A satisfies ψ[a, b, c, d] if and only if d is the greatest common divisor of a, b, and c. 11. Let L be a language and let ϕ be an L-sentence. The spectrum of ϕ is the set of cardinalities of finite models of ϕ, i.e. it is the set of all natural numbers n for which ϕ is satisfied in some structure the underlying set of which has exactly n elements. a For each of the following subsets of N, exhibit, when it is possible, an example of a language L and an L-sentence ϕ whose spectrum is the subset in question. (a) ; (b) N; (c) Z + ; (d) {n Z + ( p N)(n = 2p)} 4

5 (e) {n Z + ( p N)(n = p 2 )}; (f) {3}; (g) {1, 2, 3, 4}; (h) N \ {0, 1,..., k}; (i) the set of non-zero composite natural numbers; (j) the set of prime numbers. b Show that any sentence whose spectrum is infinite has at least one infinite model. 5

CS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati

CS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati Important 1. No questions about the paper will be entertained during