Set Proof
Empty set and power set – proof or counterexample?
Prove or provide a counterexample to disprove the assertion: (let 0 denote the empty set)
A = 0 <-> P(A) = 0
Isn’t the power set of the empty set {0}? It’s not the empty set, its a set that only contains the empty set, right? So this is false?
You are correct. The powerset of an empty set is the set { ∅ }
One way to “prove” this is as follows: We know that if a set A = ∅ then that set has exactly 0 items. Also remember that given a set A with n elements, by definition, the powerset of A contains exactly 2ⁿ elements. Combining these two definitions, we see that the powerset of an empty set must be 1.
If you want a more ‘mathematical’ proof, take a look at Cantor’s theorem, which tells us that given any set A, that the cardinality of the power set of A must always be strictly greater than the cardinality of A.
Specifically, in this case, we can use it to show that if A = ∅ then P(A) ≠ ∅ since |P(A)| must be strictly greater than |A|.
Hudson Mohawke playing an exclusive Hip Hop set at the Boiler Room #21 Livin’ Proof Takeover
|
|
1996 S Silver Proof Set with COA and OGP $0.99 |
|
|
1969-pds US mint & PROOF SETS 15 COINS (2) silver kennedyS $17.99 |
|
|
1968-pds US mint & PROOF SETS 15 COINS (2) silver kennedyS $17.99 |
|
|
1964 Proof Set $27.49 |
|
|
2012 S Proof Roosevelt Dime Ten-Cent Coin 10c from US Mint Proof Set $6.95 |
|
|
1987 Proof Set Fast FREE Shipping 5 Coins Total Best Prices On Ebay WoW $6.44 |
|
|
1995 S Silver Proof Set with COA and OGP $0.99 |
|
|
1986 Proof Set Fast FREE Shipping 5 Coins Total Best Prices On Ebay WoW $7.06 |
|
|
1966 US SPECIAL MINT SET PROOF LIKE COINS $10.99 |
|
|
UNITED STATES MINT SILVER PROOF SET 2000 $31.99 |
|
|
Lot of FIVE US Mint Proof Sets, 1960-1964 w/ CoAs $119.99 |
|
|
2008 US Mint SILVER Proof Set w/ CoA $54.99 |
|
|
2007 14-piece US Mint SILVER Proof Set w/ CoA $49.99 |
|
|
2006 US Mint SILVER Proof Set w/ CoA $39.99 |
|
|
2008 US MINT 5 PIECE STATE QUARTER CLAD PROOF SET with box and COA $39.99 |
|
|
2005 US Mint SILVER Proof Set w/ CoA $44.99 |
|
|
2004 US Mint State Quarters SILVER Proof Set w/ CoA $44.99 |
|
|
2004 US Mint SILVER Proof Set w/ CoA $44.99 |
|
|
2003 US Mint SILVER Proof Set w/ CoA $39.99 |
|
|
1990 U. S. Mint Prestige Proof Set, Box and COA $20.00 |
|
|
2002 US Mint SILVER Proof Set w/ CoA $39.99 |
|
|
2001 US Mint SILVER Proof Set w/ CoA $39.99 |
|
|
2000 US Mint SILVER Proof Set w/ CoA $39.99 |
|
|
1999 US Mint SILVER Proof Set w/ CoA $99.99 |
|
|
1958 US Mint Proof Set $24.99 |
|
|
1964 U.S. Proof Set~Philadelphia Mint $0.99 |
|
|
1994 S Silver Proof Set with COA and OGP $0.99 |
|
|
Lot of US Mint Premier Silver Proof Sets $220.00 |
|
|
1983-S Roosevelt Dime UNCIRCULATED Coin from a Proof Set $1.50 |
|
|
1993-S Roosevelt Dime UNCIRCULATED Coin from a Proof Set $5.00 |
|
|
1964 U.S. Proof Set~Capital Holder $0.99 |
|
|
2012 S Presidential Proof Set ~ With Box and COA $26.95 |
|
|
1991-S Roosevelt Dime UNCIRCULATED Coin from a Proof Set $3.00 |
|
|
2002-S Roosevelt Dime UNCIRCULATED Coin from a Proof Set $2.00 |
|
|
2009-S SILVER U. S. PROOF SET $59.95 |
|
|
1964 U.S. Proof Set~Philadelphia Mint $0.99 |
|
|
1998 United States Mint Silver Proof Set $0.99 |
|
|
1998 United States Mint Silver Proof Set $0.99 |
|
|
1981 United States Proof Set In Original Holder , Super $8.44 |
|
|
Lot of 23 Original Mint Sealed Envelopes 1956 Silver Proof Sets Unopened $1,349.99 |
|
|
Proof $4.5 Top-notch forensic scientist Alexandra Forsythe returned to Athena to prove that the death of her dearest friend had been no accident. Armed with only her razor-sharp mind and coolness under fire–and the memory of a desperate call for help–Alex set out to uncover a truth that could shake the foundations of the academy that had trained her. Her digging provoked deadly retaliation and the attentions of a stranger who might lead her toward the truth–or her death. Because in the race for final proof, only the most determined would survive…. |
