Equivalence
We have already introduced the idea of equivalence
in discussing translation.
Equivalence
We have already introduced the idea of equivalence
in discussing translation.
-(AvB) is equivalent to -A&-B.
Equivalence
We have already introduced the idea of equivalence
in discussing translation.
-(AvB) is equivalent to -A&-B.
A translation exercise is correct if it is equivalent to
an answer I would have chosen.
Equivalence
A is equivalent to B
iff
A entails B and B entails A.
Testing for Equivalence
To show A is equivalent to B ...
with a table:
The output rows match.
with a proof:
Given A, prove B,
and given B, prove A.
with a tree:
The tree for A, -B closes
and the tree for B, -A closes.
Examples
To show -(PvQ) is equivalent to -P&-Q ...
with a table:
The output rows match.
with a proof:
Given -(PvQ), prove -P&-Q,
and given -P&-Q, prove -(PvQ).
with a tree:
The tree for -(PvQ), -(-P&-Q) closes
and the tree for -P&-Q, --(PvQ) closes.
Examples
To show -(PvQ) is equivalent to -P&-Q ...
with a table:
P
T
F
T
F
Q
T
T
F
F
- (P v Q)
F T
F T
F T
T F
*
The output rows match.
-P & -Q
F F F
T F F
F F T
T T T
*
Examples
To show -(PvQ) is equivalent to -P&-Q ...
with a table:
with a proof:
The output rows match.
Given -(PvQ), prove -P&-Q,
and given -P&-Q, prove -(PvQ).
Examples
To show -(PvQ) is equivalent to -P&-Q ...
with a table:
with a proof:
1) -(PvQ)
2) -P&-Q
A
1 DM
The output rows match.
Given -(PvQ), prove -P&-Q,
and given -P&-Q, prove -(PvQ).
1) -P&-Q
2) -(PvQ)
A
1 DM
Examples
To show -(PvQ) is equivalent to -P&-Q ...
with a table:
The output rows match.
with a proof:
Given -(PvQ), prove -P&-Q,
and given -P&-Q, prove -(PvQ).
with a tree:
The tree for -(PvQ), -(-P&-Q) closes
and the tree for -P&-Q, --(PvQ) closes.
Examples
To show -(PvQ) is equivalent to -P&-Q ...
with a table:
The output rows match.
with a proof:
Given -(PvQ), prove -P&-Q,
and given -P&-Q, prove -(PvQ).
with a tree:
-(PvQ)
-(-P&-Q)
-P
-Q
--P
*
--Q
*
The tree for -(PvQ), -(-P&-Q) closes
and the tree for -P&-Q, --(PvQ) closes.
-P&-Q
--(PvQ)
-P
-Q
PvQ
P
*
Q
*
Summary of the Tests
To show A is equivalent to B ...
with a table:
The output rows match.
with a proof:
Given A, prove B,
and given B, prove A.
with a tree:
The tree for A, -B closes
and the tree for B, -A closes.
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