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Think of 'not' as being an inverse. Not 1 = 0. Not 0 = 1. Using boolean algebra we can look at your question.
'and' is a test. It wants to know if BOTH P and Q are the same and if they are 1 (true). If they are not the same, or they are both 0, then the result is false or 0.
not P and Q is rewritten like so: (P and Q)' = X not P and not Q is rewritten like: P' and Q' = X (the apostrophe is used for not)
We will construct a truth table for each and compare the output. If the output is the same, then you have found your equivalency. Otherwise, they are not equivalent. P and Q are the inputs and X is the output.
P Q | X P Q | X
------ 0 0 | 1 0 0 | 1 0 1 | 1 0 1 | 0 1 0 | 1 1 0 | 0 1 1 | 0 1 1 | 0
Since the truth tables are not equal, not P and Q is not equivalent to not P and not Q.
Perhaps you meant "Is NOT(P AND Q) equivalent to NOT(P) AND NOT(Q)?"
NOT(P AND Q) can be thought of intuitively as "Not both P and Q." Which if you think about, you can see that it would be true if something were P but not Q, Q but not P, and neither P nor Q-- so long as they're not both true at the same time.
Now, "NOT(P) AND NOT(Q)" is clearly _only_ true when BOTH P and Q are false. So there are cases where NOT(P AND Q) is true but NOT(P) AND NOT(Q) is false (an example would be True(P) and False(Q)).
NOT(P AND Q) does have an equivalence however, according to De Morgan's Law. The NOT can be distributed, but in doing so we have to change the "AND" to an "OR".
NOT(P AND Q) is equivalent to NOT(P) OR NOT(Q)
What else can I help you with?
How does each multiplication problem compare to its cooresponding division problem?
By inverting the numbers. For example, if:2 x 3 = 6 then: 6 / 3 = 2
When can you say that the given number is a rational?
A number is said to be rational if it can be expressed as a ratio of two integers. That is, a number x is rational if and only if it is equivalent to p/q for some integers p and q where q is not 0.
What does p over q mean in algebra?
P! / q!(p-q)!
What is 4 times the sum of q and p?
4(p + q), or 4p + 4q
What are the inference rules for functional dependency?
"The present list of 19 rules of inference constitutes a COMPLETE system of truth-functional logic, in the sense that it permits the construction of a formal proof of validity for ANY valid truth-functional argument." (FN1)The first nine rules of the list are rules of inference that "correspond to elementary argument forms whose validity is easily established by truth tables." (Id, page 351). The remaining ten rules are the Rules of Replacement, "which permits us to infer from any statement the result of replacing any component of that statement by any other statement logically equivalent to the component replaced." (Id, page 359).Here are the 19 Rules of Inference:1. Modus Ponens (M.P.)p qpq 2.Modus Tollens (M.T.)p q~q~p 3.Hypothetical Syllogism (H.S.)p qq rp r 4.Disjunctive Syllogism (D.S.)p v q~ pq 5. Constructive Dilemma (C.D.)(p q) . (r s)p v rq v s 6. Absorption (Abs.)p qp (p. q)7. Simplification (Simp.)p . qp 8. Conjunction (Conj.)pqp . q 9. Addition (Add.)pp v qAny of the following logically equivalent expressions can replace each other wherever they occur:10.De Morgan's Theorem (De M.) ~(p . q) (~p v ~q)~(p v q) (~p . ~q) 11. Commutation (Com.)(p v q) (q v p)(p . q) (q . p) 12. Association (Assoc.)[p v (q v r)] [(p v q) v r][p . (q . r)] [(p . q) . r] 13.Distribution (Dist) [p . (q v r)] [(p . q) v (p . r)][p v (q . r)] [(p v q) . (p v r)] 14.Double Negation (D.N.)p ~ ~p 15. Transposition (Trans.)(p q) (~q ~p) 16. Material Implication (M. Imp.)(p q) (~p v q) 17. Material Equivalence (M. Equiv.)(p q) [(p q) . (q p)](p q) [(p . q) v (~p . ~q)] 18. Exportation (Exp.)[(p . q) r] [p (q r)] 19. Tautology (Taut.) p (p v p)p (p . p)FN1: Introduction to Logic, Irving M. Copi and Carl Cohen, Prentice Hall, Eleventh Edition, 2001, page 361. The book contains the following footnote after this paragraph: "A method of proving this kind of completeness for a set of rules of inference can be found in I. M. Copi, Symbolic Logic, 5th Edition. (New York: Macmillian, 1979), chap 8, See also John A. Winnie, "The Completeness of Copi's System of Natural Deduction," Notre Dame Journal of Formal Logic 11 (July 1970), 379-382."
What statement is logically equivalent to "If p, then q"?
The statement "If not q, then not p" is logically equivalent to "If p, then q."
P-q and q-p are logically equivalent prove?
p --> q and q --> p are not equivalent p --> q and q --> (not)p are equivalent The truth table shows this. pq p --> q q -->(not)p f f t t f t t t t f f f t t t t
Wha tis the equivalent fractions of two numbers?
If the two numbers are p and q (where q is non zero) then the equivalent fraction is p/q.
What is the equivalent of an inverse statement?
The equivalent of an inverse statement is formed by negating both the hypothesis and the conclusion of a conditional statement. For example, if the original statement is "If P, then Q" (P → Q), the inverse would be "If not P, then not Q" (¬P → ¬Q). While the inverse is related to the original statement, it is not necessarily logically equivalent.
Is not(p and q) equal to (not p) or q?
No, the statement "not(p and q)" is not equal to "(not p) or q." According to De Morgan's laws, "not(p and q)" is equivalent to "not p or not q." This means that if either p is false or q is false (or both), the expression "not(p and q)" will be true. Therefore, the two expressions represent different logical conditions.
Are there equivalent numbers?
Yes. For example, the rational number p/q is equivalent to the number (a*p)/(a*q) for any non-zero number a.
What is the equivalent ratio?
Given one ratio, p/q, you will obtain an equivalent ratio if you multiply p and q by any non-zero integer.
How do you rewrite a biconditional as two conditional statements?
A biconditional statement, expressed as "P if and only if Q" (P ↔ Q), can be rewritten as two conditional statements: "If P, then Q" (P → Q) and "If Q, then P" (Q → P). This means that both conditions must be true for the biconditional to hold. Essentially, the biconditional asserts that P and Q are equivalent in truth value.
How can the statement "p implies q" be expressed in an equivalent form using the logical operator "or" and the negation of "p"?
The statement "p implies q" can be expressed as "not p or q" using the logical operator "or" and the negation of "p".
What is equivalent to the expression p 4 p 12 q?
The expression ( p^4 \cdot p^{12} \cdot q ) can be simplified using the property of exponents that states ( a^m \cdot a^n = a^{m+n} ). Therefore, ( p^4 \cdot p^{12} = p^{4+12} = p^{16} ). The equivalent expression is ( p^{16} q ).
What are the example of equivalent sets?
P={1,3,5,7,9} q={2,4,6,8,10}
What is P and q implies not not p or r if and only if q?
The statement "P and Q implies not not P or R if and only if Q" can be expressed in logical terms as ( (P \land Q) \implies (\neg \neg P \lor R) \iff Q ). This can be simplified, as (\neg \neg P) is equivalent to (P), leading to ( (P \land Q) \implies (P \lor R) \iff Q ). The implication essentially states that if both (P) and (Q) are true, then either (P) or (R) must also hold true, and this equivalence holds true only if (Q) is true. The overall expression reflects a relationship between the truth values of (P), (Q), and (R).
