Question

(1) For each assertion below, indicate if it is true (T) or false (F) by circling the correct response. (a) (T, F) The statement, "this statement is false” is a proposition. (b) (T, F) The statement "If I am Spider-Man, then I can breath in space" is true. (c) (TF) The statement "Spider-Man can breath in space" is true. (d) (T. F) "F →p can be false. (e) (T,F) "The proposition p p is not satisfiable. (f) (T, F) The proposition p q has the same truth value as p. (g) (T. F) p q + p V q is a tautology. (h) (T, F) Vr(P ) AQ()) +VrP:) A VEQ(x) is a tautology. (i) (T. F) Ar(P ) AQ(x)) + 30P(x) A 3rQ(x) is a tautology. 6) (T, F) If U =Z+ then Vnm(n? <m) is true. (k) (T. F) If U = Z+ then Im Vn (n? <m) is true. (1) (T. F) If U =R then Ir(0<r <1). (m) (T, F) If U = Z then r(0<r <1). (n) (T, F) #{} (o) (T, F) {0,1} € {0,{1}}. (p) (T. F) {0,1} = {0,{0,1}}. (q) (T. F) Ø¢{{@}} (r) (T, F) {a,b} c{a, a, b}. (s) (T, F) {a} € {a,b, {a,b}}. (t) (T. F) 6 € {a, {a,b}}.

Please answer the problems after f and please explain the reasoning

Answer 1

(a) T

This statement is indeed a proposition, however it is always False.

(b) T

This is statement is vacuously true for anyone who is not Spiderman, and if the person claiming this is indeed Spiderman then it is true since he can breath in space.

(c) T

This is true, since Spiderman was able to breath in Titan.

(d) F

The given statement can never be false since it means that if False is true then p will hold. Since False is never true, whatever be the value of p, the statement is never false.

(e) T

This proposition is satisfiable when both p and the negation of p are simultaneously true. But since that can never happen it is unsatisfiable.

(f) T

The first proposition means whenever p is true, q must be true. This in turn means that q not being true implies p is also not true.

(g) T

First we will prove the forward direction. p implies q means that if p is true then q must be true. So q is false only when p is false, which means the negation of p is true. So either the negation of p is true or q is true.
We will prove the backward direction that either the negation of p is true or q is true. This means either p is false or q is true. Since either of the negation of p and q must be true, both cannot be false simultaneously. So p cannot be false and q be true at the same time. Thus the negation of p being true implies q is true.

(h) T

First we prove the forward direction. Since for all x, P(x) and Q(x) are true simultaneously we get for all x P(x) is true and for all x Q(x) is true.
We now prove the reverse direction. If P(x) is true for all x and so is Q(x) then for all x P(x) and Q(x) is true.

(i) F

The reverse direction of this proposition is not true. There exists a x for which P(x) is true and there exists another x for which Q(x) is true does not necessarily mean that there must be a x for which P(x) and Q(x) is simultaneously true.

(j) T

Since the set of positive integers is unbounded above, for any value of n we will get a m which is greater than the square of n. For example we can have m=n²+1.

(k) F

There is no positive integer m such that m is greater than the square of all positive integers.

(l) T

0.2 is a possible value of x.

(m) F

There is no integer greater than 0 and less than 1.

(n) F

Both represent the empty or null set.

(o) F

The set {0,1} does not belong to the set {0, {1}}

(p) F

The set {0,1} is not a subset of {0, {0,1}} but an element.

(q) T

The null set does not belong to the set of set of null set.

(r) T

Since each element of {a,b} belong to {a,a,b} the former is a subset of the later.

(s) F

The set {a} does not belong to {a,b,{a.b}}

(t) F

b does not belong to {a,{a,b}}