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Solved Examples on Symbolic Logic: Predicate Logic into Symbols

Solve all questions.
Show all work.
You may use any of the Laws of Logical Equivalences and/or Logical Equivalences for Propositional Logic and Predicate Logic.
For any law of logical equivalence that you use, please indicate the law or the logical equivalence number.

(1.) Statement: The reciprocal of every negative real number is also negative.
Translate this statement into symbolic logic.
Determine the truth value.

$ Let\;\;x\;\;be\;\;a\;\;negative\;\;number \\[3ex] Every\;\;negative\;\;number = \forall x \\[3ex] Reciprocal\;\;of\;\;every\;\;negative\;\;number = \dfrac{1}{x} \\[5ex] Negative\;\;means\;\; \lt 0 \\[3ex] \underline{Symbolic\;\;Logic} \\[3ex] \forall x \lt 0 \left(\dfrac{1}{x} \lt 0\right) \\[5ex] The\;\;statement\;\;is\;\;true \\[3ex] The\;\;truth\;\;value = T $

(2.)

$ LHS \\[3ex] \neg[\forall x (p(x) \land q(x))] \\[3ex] \equiv \neg \forall x \; \neg(p(x) \land q(x)) \\[3ex] *** \\[3ex] \neg \forall x \equiv \exists x ...Number(11.)...De\:\:Morgan's\:\:Law \\[3ex] \neg(p(x) \land q(x)) \equiv \neg p(x) \lor \neg q(x) ...Number(2.)...De\:\:Morgan's\:\:Law \\[3ex] *** \\[3ex] \equiv \exists x(\neg p(x) \lor \neg q(x)) \\[3ex] = RHS $

For Questions (3.) through (8.):
The domain of discourse is the set of all SNHU students.
Given the logical statements:
$p(x)$: $x$ is an scholar.
$q(x)$: $x$ likes Discrete Mathematics.
$r(x)$: $x$ plays basketball.

(I.) Determine the truth value of each English logical statement.
(II.) Translate each statement into symbolic logic.
(III.) Negate it.
(IV.) Apply De Morgan's Law
(V.) Translate the symbolic logic back to English language.
(VI.) Determine the truth value of the negation.

(3.) Every SNHU student plays basketball or likes Discrete Mathematics or both.

Every SNHU student plays basketball or likes Discrete Mathematics or both.

(I.) This is false.
The truth value is $F$.

$ (II.) \\[3ex] \forall x(r(x) \lor q(x)) \\[3ex] (III.) \\[3ex] Negate\;\;it \\[3ex] \neg [\forall x(r(x) \lor q(x))] \\[3ex] (IV.) \\[3ex] \neg [\forall x(r(x) \lor q(x))] \\[3ex] \equiv \neg \forall x \; \neg(r(x) \lor q(x)) \\[3ex] De'\;Morgan's\;\;Law \\[3ex] *** \\[3ex] \neg \forall x \equiv \exists x ...No.(11.)\;De\:\:Morgan's\:\:Law \\[3ex] \neg(r(x) \lor q(x)) \equiv \neg r(x) \land \neg q(x) ...No.(1.)\;De\:\:Morgan's\:\:Law \\[3ex] *** \\[3ex] \equiv \exists x (\neg r(x) \land \neg q(x)) \\[3ex] $ (V.) Some SNHU students do not play basketball and do not like Discrete Mathematics.

(VI.) This is true.
The truth value is $T$

(4.) There exists at least one SNHU student who plays basketball and likes Discrete Mathematics.

There is at least an SNHU student who plays basketball and likes Discrete Mathematics.

(I.) This is 'most likely' true.
The truth value is $T$.

$ (II.) \\[3ex] \exists x(r(x) \land q(x)) \\[3ex] (III.) \\[3ex] Negate\;\;it \\[3ex] \neg [\exists x(r(x) \land q(x))] \\[3ex] (IV.) \\[3ex] \neg [\exists x(r(x) \land q(x))] \\[3ex] \equiv \neg \exists x \; \neg(r(x) \land q(x)) \\[3ex] De'\;Morgan's\;\;Law \\[3ex] *** \\[3ex] \neg \exists x \equiv \forall x ...No.(12.)\;De\:\:Morgan's\:\:Law \\[3ex] \neg(r(x) \land q(x)) \equiv \neg r(x) \lor \neg q(x) ...No.(2.)\;De\:\:Morgan's\:\:Law \\[3ex] *** \\[3ex] \equiv \forall x (\neg r(x) \lor \neg q(x)) \\[3ex] $ (V.) Every SNHU student either does not play basketball or does not like Discrete Mathematics or both.

(VI.) This is false.
The truth value is $F$

(5.) Every SNHU student who is a scholar likes Discrete Mathematics.

Every SNHU student who is a scholar likes Discrete Mathematics.

(I.) This is 'most likely' not true.
The truth value is $F$.

$ (II.) \\[3ex] \forall x(p(x) \rightarrow q(x)) \\[3ex] (III.) \\[3ex] Negate\;\;it \\[3ex] \neg [\forall x(p(x) \rightarrow q(x))] \\[3ex] (IV.) \\[3ex] \neg [\forall x(p(x) \rightarrow q(x))] \\[3ex] \equiv \neg \forall x \; \neg(p(x) \rightarrow q(x)) \\[3ex] *** \\[3ex] \neg \forall x \equiv \exists x ...No.(11.)\;De\:\:Morgan's\:\:Law \\[3ex] \neg(p(x) \rightarrow q(x)) \equiv p(x) \land \neg q(x)...No.(7.) \\[3ex] *** \\[3ex] \equiv \exists x (p(x) \land \neg q(x)) \\[3ex] $ (V.) There exists one SNHU student who plays basketball and does not like Discrete Mathematics.

(VI.) This is 'most likely' true.
The truth value is $T$

(6.)

$ RHS \\[3ex] \neg[\forall x (\neg p(x) \land q(x))] \\[3ex] \equiv \neg \forall x \; \neg(\neg p(x) \land q(x)) \\[3ex] *** \\[3ex] \neg \forall x \equiv \exists x ...Number(11.)...De\:\:Morgan's\:\:Law \\[3ex] \neg(\neg p(x) \land q(x)) \equiv \neg \neg p(x) \lor \neg q(x) ...Number(2.)...De\:\:Morgan's\:\:Law \\[3ex] \neg \neg p(x) \equiv p(x) ...Number(15.)...Double\:\:Negation\:\:Law \\[3ex] *** \\[3ex] \equiv \exists x(p(x) \lor \neg q(x)) \\[3ex] = LHS $

(7.)

$ RHS \\[3ex] \neg[\exists x (\neg p(x) \rightarrow q(x))] \\[3ex] \equiv \neg \exists x \; \neg(\neg p(x) \rightarrow q(x)) \\[3ex] *** \\[3ex] \neg \exists x \equiv \forall x ...Number(12.)...De\:\:Morgan's\:\:Law \\[3ex] \neg p(x) \rightarrow q(x) \equiv p(x) \lor q(x) ...Number(4.) \\[3ex] \therefore \neg(\neg p(x) \rightarrow q(x)) \equiv \neg (p(x) \lor q(x)) \\[3ex] \neg (p(x) \lor q(x)) \equiv \neg p(x) \land \neg q(x) ...Number(1.)...De\:\:Morgan's\:\:Law \\[3ex] *** \\[3ex] \equiv \forall x(\neg p(x) \land \neg q(x)) \\[3ex] = LHS $

(8.)

$ LHS \\[3ex] \neg[\forall x (p(x) \rightarrow \neg q(x))] \\[3ex] \equiv \neg \forall x \; \neg(p(x) \rightarrow \neg q(x)) \\[3ex] *** \\[3ex] \neg \forall x \equiv \exists x ...Number(11.)...De\:\:Morgan's\:\:Law \\[3ex] p(x) \rightarrow \neg q(x) \equiv \neg p(x) \lor \neg q(x) ...Number(5.) \\[3ex] \therefore \neg(p(x) \rightarrow \neg q(x)) \equiv \neg(\neg p(x) \lor \neg q(x)) \\[3ex] \neg(\neg p(x) \lor \neg q(x)) \equiv \neg \neg p(x) \land \neg \neg q(x) ...Number(1.)...De\:\:Morgan's\:\:Law \\[3ex] \neg \neg p(x) \equiv p(x) ...Number(15.)...Double\:\:Negation\:\:Law \\[3ex] \therefore \neg \neg p(x) \land \neg \neg q(x) \equiv p(x) \land q(x) \\[3ex] *** \\[3ex] \equiv \exists x(p(x) \land q(x)) \\[3ex] = RHS $

(9.) Statement: The reciprocal of every positive number less than or equal to one is greater than or equal to one.
Translate this statement into symbolic logic.
Determine the truth value.

$ Let\;\;x\;\;be\;\;a\;\;positive\;\;number \\[3ex] Every\;\;positive\;\;number = \forall x \\[3ex] Positive\;\;means\;\; \gt 0 \\[3ex] Less\;\;than\;\;or\;\;equal\;\;to\;\;1\;\; means \;\; \le 1 \\[3ex] Every\;\;positive\;\;number\;\;less\;\;than\;\;or\;\;equal\;\;to;\;one = \forall x((x \gt 0) \;\;and\;\; (x \le 1)) \\[3ex] Reciprocal\;\;of\;\;every\;\;positive\;\;number = \dfrac{1}{x} \\[5ex] Greater\;\;than\;\;or\;\;equal\;\;to\;\;1\;\; means \;\; \ge 1 \\[3ex] \underline{Symbolic\;\;Logic} \\[3ex] \forall x[(x \gt 0) \;\;\land\;\; (x \le 1)] \rightarrow \dfrac{1}{x} \ge 1 \\[5ex] Example:\;\; x = 0.5 \\[3ex] Because:\;\; 0.5 \gt 0 \;\;and\;\; 0.5 \le 1 \\[3ex] \dfrac{1}{0.5} = 2 \\[5ex] 2 \ge 1 \\[3ex] No\;\; counter\;\;example \\[3ex] The\;\;statement\;\;is\;\;true \\[3ex] The\;\;truth\;\;value = T $

(10.) Statement: There are two real numbers whose ratio is greater than three.
Translate this statement into symbolic logic.
Determine the truth value.

$ Let\;\;the\;\;two\;\;numbers\;\;be\;\;x\;\;and\;\;y \\[3ex] \exists x \; \exists y \left(\dfrac{x}{y} \gt 3\right) \\[5ex] Example:\;\; x = 7, y = 2 \\[3ex] \dfrac{7}{2} \gt 3 \\[5ex] The\;\;statement\;\;is\;\;true \\[3ex] The\;\;truth\;\;value = T $

(11.) There is no greatest number.

There is no greatest number
This means that for any number say $x$, there exists another number say $y$ such that $y$ is greater than $x$

$ \forall x \; \exists y\;(y \gt x) $