Math Problem Statement
Solution
Let's go through the truth tables one by one and solve the questions based on the image.
Part a - Completing the truth table:
You are asked to fill out the truth table for p
, ~p
, and ~(~p)
.
p | ~p | ~(~p) |
---|---|---|
T | F | T |
F | T | F |
- The column for
~p
is the negation ofp
. - The column for
~(~p)
is the negation of~p
, which makes it equivalent top
.
Part b - Completing the truth table:
This part involves filling in the truth table for logical disjunction (∨
) and conjunction (∧
) operations:
p | ~p | p ∨ ~p | p ∧ ~p |
---|---|---|---|
T | F | T | F |
F | T | T | F |
p ∨ ~p
: This is the disjunction (OR) betweenp
and~p
. It results in true whenever at least one operand is true.p ∧ ~p
: This is the conjunction (AND) betweenp
and~p
. It results in false becausep
and~p
are always opposites, and both can't be true at the same time.
Part c - Is p logically equivalent to ~(~p)
?
Yes, based on the truth table in part (a), p
is logically equivalent to ~(~p)
. Both columns for p
and ~(~p)
are identical, showing the same truth values.
Part d - Is p ∨ ~p
logically equivalent to p ∧ ~p
?
No, based on the truth table in part (b), p ∨ ~p
is not logically equivalent to p ∧ ~p
. The truth values for p ∨ ~p
are always true, while the truth values for p ∧ ~p
are always false.
Would you like more details or further explanation on any part? Here are five related questions to expand on this topic:
- What is the difference between a tautology and a contradiction in logic?
- How would the truth table change if we introduce a third logical operator, such as XOR (exclusive OR)?
- Can you explain the De Morgan's laws based on these truth tables?
- How do truth tables help in proving logical equivalence in Boolean algebra?
- How can we use these truth tables to evaluate more complex logical expressions?
Tip: In logic, an expression like p ∨ ~p
is a tautology (always true), while p ∧ ~p
is a contradiction (always false).
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Math Problem Analysis
Mathematical Concepts
Logic
Boolean Algebra
Formulas
Negation: ~p
Conjunction: p ∧ q
Disjunction: p ∨ q
Double Negation: ~(~p)
Theorems
Law of Double Negation
Law of Excluded Middle
Law of Noncontradiction
Suitable Grade Level
College
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