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It is the most powerful prayer. A pure heart, a clean mind, and a clear conscience is necessary for it. - Samuel Dominic Chukwuemeka

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Solved Examples on Symbolic Logic: Propositional Logic into Symbols

For ACT Students
The ACT is a timed exam...$60$ questions for $60$ minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for a wrong answer.

For WASSCE Students
Any question labeled WASCCE is a question for the WASCCE General Mathematics
Any question labeled WASSCE:FM is a question for the WASSCE Further Mathematics/Elective Mathematics

Solve all questions.
Show all work.

(1.) WASSCE Consider the statements:
p: it is hot.
q: it is raining.
Which of the following symbols correctly represents the statement "It is raining if and only if it is cold"?

$ A.\;\; p \leftrightarrow \neg q \\[3ex] B.\;\; q \leftrightarrow p \\[3ex] C.\;\; \neg p \leftrightarrow \neg q \\[3ex] D.\;\; q \leftrightarrow \neg p \\[3ex] $

p: It is hot
not p: It is cold.
q: it is raining.

It is raining if and only if it is cold
q if and only if not p

$ q \leftrightarrow \neg p $

(2.) WASCCE If p = Musa is short,
q = Musa is brilliant,
write, in symbolic form, the statement "Musa is short but not brilliant."

$ A.\;\; p \lor q \\[3ex] B.\;\; p \lor \neg q \\[3ex] C.\;\; p \land \neg q \\[3ex] D.\;\; p \land q \\[3ex] $

p = Musa is short
q = Musa is brilliant
not q = Musa is not brilliant.
In logic, but means and

Musa is short but not brilliant.
p and not q

$ p \land \neg q $

(4.)

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.

(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.)

$ 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.)

$ 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 $