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 $