Given the statements below, write the following symbolic compound statements in sentences.
Statement \(p\text{:}\) The car will not start.
Statement \(q\text{:}\) It is freezing outside.
Statement \(r\text{:}\) The food is ready.
\(p \land q\)
\(\displaystyle q \lor (\lnot r)\)
\(\displaystyle p \land (\lnot )\)
\(\displaystyle p \land q \land r\)
Checkpoint1.4.2.
Given the statements above, write the following sentences in logical notation:
The car will start and it is freezing outside.
Either the food is ready or it is freezing outside.
The food isn’t ready and the car won’t start.
Checkpoint1.4.3.
Construct a truth table for each of the following statements.
\(p \land (\lnot q)\text{;}\)
\(p \lor (q \land (\lnot p))\text{;}\)
\(p \land (q \lor (\lnot p))\text{;}\)
\(p \land (\lnot q)\text{.}\)
Checkpoint1.4.4.
Identify the hypothesis \(p\) and the conclusion \(q\) in each of the following conditional statements.
I will go to the movies if it rains.
If the diagonals of a rectangle are not perpendicular, it is not a square.
If \(x > 0\) and \(y > 0\text{,}\) then \(x+y > 0\text{.}\)
\(f(x)\) is a function on \([a,b]\) implies that \(f(x)\) is continuous on \([a,b]\text{.}\)
I will pay 5 dollars for a cup of coffee only if elephants fly.
All sides are equal provided that \(T\) is an equilateral triangle.
Parellel opposite sides is sufficient for a quadrilateral to be a parallelogram.
Checkpoint1.4.5.
Use truth tables to prove that disjunction and conjunction are associative and that each one distributes over the other. That is, for all statements \(p\text{,}\)\(q\) and \(r\text{,}\) prove the following.
\(\displaystyle (p \land q) \land r \equiv p \land (q \land r)\)
\(\displaystyle (p \lor q) \lor r \equiv p \lor (q \lor r)\)
\(\displaystyle p \land( q \lor r) \equiv (p \land q) \lor ( p \land r)\)
\(\displaystyle p \lor( q \land r) \equiv (p \lor q) \land ( p \lor r)\)
Checkpoint1.4.6.
Is implication associative? That is, are the compound statements \(\left [(p \implies q) \implies r \right ]\) and \(\left [ p \implies(q \implies r)\right ]\) logically equivalent? Justify your answer!
Checkpoint1.4.7.
Form the contrapositive, converse, and inverse of each of the following statements in the form of an English sentence (no symbolic logic notation)
If two lines are perpendicular, then they intersect at a right angle.
Function \(f(x)\) is differentiable provided that \(f(x)\) is a polynomial.
(Divergence Test) If \(\sum_{n=1}^\infty a_n\) converges, then \(\lim_{n\to \infty} a_n = 0\text{.}\)
For all \(n\times n\) matrices \(A\) and \(B\text{,}\) if \(A\) and \(B\) are both nonsingular, then \(AB\) is nonsingular.
