Lecture 4 Questions
1. Consider all people in the world. Given persons x, and y, let L(x, y) be
the statement ‘x loves y’. Rewrite the following using formal mathematics,
such as quantifiers and implications. Also write the statement in pure logic
(using ∀, ∃, →, and L(x, y)). The solution to the first example is given.
(a) God loves everybody.
Solution: For every person p, it is the case that God loves p. ∀p, L(God, p).
(b) Everybody loves somebody.
(c) Somebody is loved by everybody.
(d) Someone loves everybody.
(e) Everybody loves themselves
(f) Narcisuss does not love anyone except himself.
2. Negate every statement from Problem 1, and state the negation in a format
similar to the original. Here is the negation for the first statement.
• Pure logic: ∃p 3 ¬L(God, p).
• Mathematical formulation: ‘There exists a person p such that God
does not love p.’
• Informal statement: ‘There is someone that God does not love’.
Definition 1. Consider two statements P , and Q.
(a) The converse of P → Q is Q → P (the order changes).
(b) The contrapositive of P → Q is ¬Q → ¬P (the order changes, and
both statements are negated).
(c) The statement P ↔ Q, written as ‘P if and only if Q’, is a shorthand
for the statement ‘(P → Q) ∧ (Q → P )’. In other words, P ↔ Q is
true when P and Q have the same truth value, and is false when P
and Q have different truth values.
3. Write down a truth table for P → Q, the converse, and the contrapositive.
Do you notice any similarities?
4. Find the converse of the following statement: If a function f (x) is continuous at x = c, then f (x) is differentiable at x = c. Bonus: Is the
statement true? If not, is the converse true? What is the contrapositive
of the statement?
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5. Which of the following are true? Justify your answers.
(a) If the Aggies are the best team in the PAC 12, then Dr. White is
going to become the next president.
(b) Given integers m and n, m and n are both even if and only if mn is
even.
(c) Given an integer n, n is divisible by 5 if and only if the ones places
is either 0 or 5.
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