Question

Prove that (A-B) ∪ (B-A) = (A∪B) - (A∩B) using propositional logic and definitions of set operators. Please state justification for each step!

Answer #1

5) Use propositional logic to prove that the argument is
valid.
A’ Λ (B → A) → B’
1.____________ ____________
2.____________ ____________
3.____________ ____________

Use the laws of propositional logic to prove the following:
(p ∧ q) → r ≡ (p ∧ ¬r) → ¬q

Prove the statements (a) and (b) using a set element proof and
using only the definitions of the set operations (set equality,
subset, intersection, union, complement):
(a) Suppose that A ⊆ B. Then for every set C, C\B ⊆ C\A.
(b) For all sets A and B, it holds that A′ ∩(A∪B) = A′ ∩B.
(c) Now prove the statement from part (b)

Philosophy
3. Multiple-Line Truth Functions
Compound statements in propositional logic are truth functional,
which means that their truth values are determined by the truth
values of their statement components. Because of this truth
functionality, it is possible to compute the truth value of a
compound proposition from a set of initial truth values for the
simple statement components that make up the compound statement,
combined with the truth table definitions of the five propositional
operators.
To compute the truth value...

Use the laws of propositional logic to prove the following:
1) (p ∧ q ∧ ¬r) ∨ (p ∧ ¬q ∧ ¬r) ≡ p ∧ ¬r
2) (p ∧ q) → r ≡ (p ∧ ¬r) → ¬q

#1. Use propositional logic to prove the following
argument is valid.
If Alice gets the office position and works hard, then
she will get a bonus. If she gets a bonus, then she will go on a
trip. She did not go on a trip. Therefore, either she did not get
the office position or she did not work hard or she was late too
many times.
Define your propositions [5 points]:
O =
W =
B =
T =...

Using predicate logic, prove De Morgan's Laws for logic.

Is the following argument valid? (Carefully express it in
propositional logic, and use the proper rules of inference at each
step. You can score well in the GMAT only if you have good
analytical skills. Every student who takes Discrete Math has good
analytical skills or good memory. Tom doesn’t have good analytical
skill. Therefore, if Tom takes Discrete Math, then Tom will score
well in the GMAT.

1)Translate all the definitions needed to derive the conclusion
into propositional formulas (i.e., premises)
2)Derive your proof for S using natural deduction with the
premises.
S: there exist irrational numbers a and b such that a^b is
rational
A: √2 is rational

1)Translate all the definitions needed to derive the conclusion
into propositional formulas (i.e., premises)
2)Derive your proof for S using natural deduction with the
premises.
S: there exist irrational numbers a and b such that a^b is
rational
A: √2 is rational
B: √2^√2 is rational
C: √2^√2*√2 is rational

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