Sufficient/Necessary Flashcards
Define Sufficient and Necessary:
A condition (X) is said to be sufficient for another condition (Y), if (and only if) the truth/existence/occurrence of X guarantees or brings about the truth/existence/occurrence of Y.
S—>N
The existence of the sufficient condition guarantees the existence of the necessary condition.
X—>Y
If we know that x exists, then we know with 100% accuracy that Y exists.
Define the Contrapositive:
If the necessary condition does not exist, then the sufficient condition cannot exist—the contrapositive.
If not N —> no S (Reversing and negating)
“IF” STATEMENTS
X—>Y
X
————
Y
(If X, then Y)
THE RULE: “If” introduces a sufficient condition. The other part of the statement is the necessary condition.
- If the batteries are dead, then the remote will not work.
-BD—>RNW
-Remote Works—>Batteries not dead.
“When” and “Whenever” Statements:
“When” and “whenever” are diagrammed just like “if”
- When there is lightening, there is thunder
-L—>Th
-Not Th—>Not L
“Where” and “wherever” Statements:
“Where” and “wherever” are diagrammed just like “if”
- Where there is smoke, there is fire
-Sm—>F
-Not F—>Not Sm
“ONLY IF” STATEMENTS:
THE RULE: “Only if” introduces a necessary condition. The other part of the statement is the sufficient condition.
- You can legally buy alcohol only if you are at least 21 years old.
-LBA—>21YO
-Not 21YO—>Can’t LBA
“Only when” Statements:
“Only when” is diagrammed just like “only if”
THE RULE: “Only if” introduces a necessary condition. The other part of the statement is the sufficient condition.
- A person can play golf only when she has a golf club.
-PG—>GC
-Don’t have GC—>Can’t PG
“Only where” Statements:
“Only where” is diagrammed just like “only if”
THE RULE: “Only if” introduces a necessary condition. The other part of the statement is the sufficient condition.
- Life can exist only where there is water
-LCanE—>W
-No W—LCannotE
“ONLY” STATEMENTS:
THE RULE: “Only” introduces a necessary condition. The other part of the statement is the sufficient condition.
- Only God can judge me.
-JM—>G
-Not G—>Can’t JM
“ONLY” v. “THE ONLY”:
THE RULE: While “only by itself always introduces a necessary condition, “the only” actually introduces a sufficient condition.
- Only vegetables are carrots.
-C—>V
-Not V—>Not C - The only vegetables are carrots.
-V—>C
-Not C—>Not V
“IF AND ONLY IF STATEMENTS”:
THE RULE: “If and only if” indicates that each variable is both a sufficient and a necessary condition. “If and only if” is diagrammed with a double arrow.
- You are a brother if and only if you are a male sibling.
-B<—>MS
-B—>MS
-Not MS—>Not B
-MS—>B
-Not B—>Not MS
“If but only if” is diagrammed just like “if and only if”: <—>
“ALL” STATEMENTS:
THE RULE: “All” introduces a sufficient condition. The other part is the necessary condition.
- All humans are mammals.
-H—>M
-Not M—> Not H
“Every” Statements:
“Every” is diagrammed like “all.”
THE RULE: “All” introduces a sufficient condition. The other part is the necessary condition.
- Every triangle has three sides.
-T—>3S
-Not 3S—>Not T
“Each” Statements:
“Each” is diagrammed just like “all.”
THE RULE: “All” introduces a sufficient condition. The other part is the necessary condition.
- Each rectangle has 4 right angles
-R—>4RA
-Not 4RA—>Not R
“Any” Statements:
“Any” is diagrammed just like “all”
THE RULE: “All” introduces a sufficient condition. The other part is the necessary condition.
- Any square has 4 sides of equal length.
-S—>4SEL
-Not 4SEL—>Not S
“PEOPLE WHO” STATEMENTS:
THE RULE: “People who” introduces a sufficient condition. The other part of the statement is the necessary condition.
- People who get pregnant are female.
-P—>F
-Not F—>Not P
“NO” STATEMENTS
The statement “no Xs are Ys” means that something cannot be both X and Y, or stated another way, X and Y are MUTUALLY EXCLUSIVE. To diagram a “no” statement:
THE RULE:
STEP 1. Pick a variable (i.e. X or Y) and make it the sufficient condition.
STEP 2. Negate the other variable and make it the necessary condition.
- No Xs are Ys
-X—>Not Y
OR
-Y—>Not X
(Contrapositives) - No mammals are cold-blooded.
-M—>Not CB
-CB—>Not M
“Not both” Statements:
“Not both” is diagrammed just like “no.”
THE RULE:
STEP 1. Pick a variable (i.e. X or Y) and make it the sufficient condition.
STEP 2. Negate the other variable and make it the necessary condition.
- One cannot be in both Los Angeles and New York at the same time.
-LA—>Not NY
-NY—>Not LA
Impossible:
LA NY
Possible:
LA, not NY
Not LA, NY
Not LA, not NY
“EITHER/OR” STATEMENTS:
The statement “either X or Y” means that AT LEAST ONE of X or Y MUST EXIST. To diagram an “either/or” statement:
THE RULE:
STEP 1. Pick a variable (i.e. X or Y) and make it the necessary condition.
STEP 2. Negate the other variable and make it the sufficient condition.
- Either X or Y
Not X—>Y
Not Y—>X - Either love me or leave me alone.
-Not LM—>LME
-Not LME—>LM
Impossible:
Not X, Not Y
Possible:
Not X, Y
X, Not Y
X, Y
Unknown:
If we only know we have Y, we can’t be certain of X’s existence or nonexistence.
“UNLESS” STATEMENTS
STEP 1. “Unless” introduces a necessary condition so the part of the sentence that follows “unless” will be the necessary condition.
STEP 2. Negate the other part of the sentence and make it the sufficient condition.
- A person cannot win the lottery unless she buys a ticket.
-WL—>BT
-Not BT—>Not WL
“Without” Statements:
“Without” is diagrammed just like “unless.”
STEP 1. “Unless” introduces a necessary condition so the part of the sentence that follows “unless” will be the necessary condition.
STEP 2. Negate the other part of the sentence and make it the sufficient condition.
- You cannot have fire without oxygen.
-F—>O
-Not O—>Not F
“Until” Statements:
“Until” is diagrammed just like “unless” and “without.”
STEP 1. “Unless” introduces a necessary condition so the part of the sentence that follows “unless” will be the necessary condition.
STEP 2. Negate the other part of the sentence and make it the sufficient condition.
- A person cannot legally vote until she is 18 years old.
-LV—>18YO or Older
-Not 18YO or Older—>Not LV
COMPOUND STATEMENTS (AND/OR):
THE RULE: Converting a compound statement into its contrapositive requires converting every “or” to “and” and every “and” to “or.”
A—> B or C
Not B & Not C —> Not A
A—>B & C
Not B or Not C—>Not A
A & B—>C
Not C—> Not A or Not C
A or B—>C
Not C–>Not A & Not B
- If one is a bachelorette, then that person must be unmarried and a female.
B—>Not M & F
M or Not F—>Not B
Must Be True:
1. Logical Opposite:
2. Logical Equivalent:
- Logical Opposite: Cannot Be False
- Logical Equivalent: Not Necessarily True
Cannot Be False:
1. Logical Opposite:
2. Logical Equivalent:
- Logical Opposite: Could Be False
- Logical Equivalent: Must Be True
Not Necessarily True:
1. Logical Opposite:
2. Logical Equivalent:
- Logical Opposite: Must Be True
- Logical Equivalent: Cannot Be False
Could Be False:
1. Logical Opposite:
2. Logical Equivalent:
- Logical Opposite: Cannot Be False
- Logical Equivalent: Not Necessarily True
Could Be True:
1. Logical Opposite:
2. Logical Equivalent:
- Logical Opposite: Cannot Be True
- Logical Equivalent: Not Necessarily False
Not Necessarily False:
1. Logical Opposite:
2. Logical Equivalent:
- Logical Opposite: Must Be False
- Logical Equivalent: Could Be True
Cannot Be True:
1. Logical Opposite:
2. Logical Equivalent:
- Logical Opposite: Could Be True
- Logical Equivalent: Must Be False
Must Be False:
1. Logical Opposite:
2. Logical Equivalent:
- Logical Opposite: Not Necessarily False
- Logical Equivalent: Cannot Be True
