Search & Filtering
. 1. Let D = {3, 9, 15}, E = {3, 15, 9}, F = {3, 3, 9, 15, 15, 15}. What are the elements of D, E, and F? How are D, E,
and F related?
2. How many elements are in the set {1, {2}, [1, 2}}?
3. For each positive integer p, let Bp = {p2, p3,]. Find B1, B2, and B3.
A home security system has a pad with 9 digit (1 to 9 ). Find the number of possible 5-digit pass code if digit can be repeated or if digit cannot be repeated
Consider the following premises:
1. A "\\to" (B "\\to" A) is a Theorem of Propositional Calculus/Logic (i.e. it’s logically valid),
for all statement forms A and B.
Suppose then that the following are the temporary axioms (assumptions):
a) W (axiom 1)
b) Y (axiom 2)
c) Y "\\to" Z (axiom 3)
Using the logical rules of inference, Modus Ponens (MP) and/or Hypothetical Syllogism
(HS), show that X "\\to" Z is deducible (i.e. it is a logical/valid conclusion) from the given
premises (i.e. 1 and 2).
a)
i) Give an inductive formula for the sum of the first n odd numbers:
1 + 3 + 5 + ... + 2n -1
Show your induction process.
ii) Use the proof by mathematical induction to prove the correctness of your
inductive formula in i) above.
What is p ⊕ p ? true or false
Let p and q be the propositions “The total amount is discounted” and “The items have been packed,” respectively. Express each of these compound propositions as an English sentence.
Construct the truth table of (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p)
Use truth tables to prove that (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p) \iff p \lor \lnot q \lor \lnot r
Construct the truth table of (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p)
Use truth tables to prove that (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p) \iff p \lor \lnot q \lor \lnot r
Explain, without using a truth table, why (p ∨ q ∨ r) ∧
(¬p ∨ ¬q ∨ ¬r) is true when at least one of p, q, and r
is true and at least one is false, but is false when all three
variables have the same truth value.
Determine whether each of these compound propositions
is satisfiable.
a) (p ∨ ¬q) ∧ (¬p ∨ q) ∧ (¬p ∨ ¬q)
b) (p → q) ∧ (p → ¬q) ∧ (¬p → q) ∧ (¬p → ¬q)
c) (p ↔ q) ∧ (¬p ↔ q)
