Mixing
What is the difference between these statements? $forall x: P(x) wedge forall y:Q(y)$ and $forall x forall...
$begingroup$
What is the difference between these statements?
$$forall x: P(x) wedge forall y:Q(y)$$
$$forall x forall y:P(x) wedge Q(y)$$
logic
edited Jan 30 at 20:23
Blue
49.4k870157
asked Jan 30 at 19:40
Ayoub RossiAyoub Rossi
11110
$endgroup$
add a comment |
$begingroup$
What is the difference between these statements?
$$forall x: P(x) wedge forall y:Q(y)$$
$$forall x forall y:P(x) wedge Q(y)$$
logic
edited Jan 30 at 20:23
Blue
49.4k870157
asked Jan 30 at 19:40
Ayoub RossiAyoub Rossi
11110
$endgroup$
2
$begingroup$
No difference...
$endgroup$
– Mauro ALLEGRANZA
Jan 30 at 19:42
2
$begingroup$
The first statement has 2 colons.
$endgroup$
– DanielV
Jan 30 at 20:59
add a comment |
$begingroup$
What is the difference between these statements?
$$forall x: P(x) wedge forall y:Q(y)$$
$$forall x forall y:P(x) wedge Q(y)$$
logic
edited Jan 30 at 20:23
Blue
49.4k870157
asked Jan 30 at 19:40
Ayoub RossiAyoub Rossi
11110
$endgroup$
What is the difference between these statements?
$$forall x: P(x) wedge forall y:Q(y)$$
$$forall x forall y:P(x) wedge Q(y)$$
logic
logic
edited Jan 30 at 20:23
Blue
49.4k870157
asked Jan 30 at 19:40
Ayoub RossiAyoub Rossi
11110
edited Jan 30 at 20:23
Blue
49.4k870157
asked Jan 30 at 19:40
Ayoub RossiAyoub Rossi
11110
edited Jan 30 at 20:23
Blue
49.4k870157
edited Jan 30 at 20:23
Blue
49.4k870157
edited Jan 30 at 20:23
Blue
49.4k870157
49.4k870157
asked Jan 30 at 19:40
Ayoub RossiAyoub Rossi
11110
asked Jan 30 at 19:40
Ayoub RossiAyoub Rossi
11110
asked Jan 30 at 19:40
Ayoub RossiAyoub Rossi
11110
11110
2
$begingroup$
No difference...
$endgroup$
– Mauro ALLEGRANZA
Jan 30 at 19:42
2
$begingroup$
The first statement has 2 colons.
$endgroup$
– DanielV
Jan 30 at 20:59
add a comment |
2
$begingroup$
No difference...
$endgroup$
– Mauro ALLEGRANZA
Jan 30 at 19:42
2
$begingroup$
The first statement has 2 colons.
$endgroup$
– DanielV
Jan 30 at 20:59
2
2
$begingroup$
No difference...
$endgroup$
– Mauro ALLEGRANZA
Jan 30 at 19:42
$begingroup$
No difference...
$endgroup$
– Mauro ALLEGRANZA
Jan 30 at 19:42
2
2
$begingroup$
The first statement has 2 colons.
$endgroup$
– DanielV
Jan 30 at 20:59
$begingroup$
The first statement has 2 colons.
$endgroup$
– DanielV
Jan 30 at 20:59
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
They are the same. The first statement asserts that $P(x)$ is true for all $x$ and that $Q(y)$ is true for all $y$.
The second statement asserts that $P(x) wedge Q(x)$ is true for all $x$ and $y$, but this thus implies that $P(x)$ is true for all $x$ and $y$ and $Q(x)$ is true for all $x$ and $y$, which is the same thing.
answered Jan 30 at 19:53
Trevor KafkaTrevor Kafka
5309
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
They are the same. The first statement asserts that $P(x)$ is true for all $x$ and that $Q(y)$ is true for all $y$.
The second statement asserts that $P(x) wedge Q(x)$ is true for all $x$ and $y$, but this thus implies that $P(x)$ is true for all $x$ and $y$ and $Q(x)$ is true for all $x$ and $y$, which is the same thing.
answered Jan 30 at 19:53
Trevor KafkaTrevor Kafka
5309
$endgroup$
add a comment |
$begingroup$
They are the same. The first statement asserts that $P(x)$ is true for all $x$ and that $Q(y)$ is true for all $y$.
The second statement asserts that $P(x) wedge Q(x)$ is true for all $x$ and $y$, but this thus implies that $P(x)$ is true for all $x$ and $y$ and $Q(x)$ is true for all $x$ and $y$, which is the same thing.
answered Jan 30 at 19:53
Trevor KafkaTrevor Kafka
5309
$endgroup$
add a comment |
$begingroup$
They are the same. The first statement asserts that $P(x)$ is true for all $x$ and that $Q(y)$ is true for all $y$.
The second statement asserts that $P(x) wedge Q(x)$ is true for all $x$ and $y$, but this thus implies that $P(x)$ is true for all $x$ and $y$ and $Q(x)$ is true for all $x$ and $y$, which is the same thing.
answered Jan 30 at 19:53
Trevor KafkaTrevor Kafka
5309
$endgroup$
They are the same. The first statement asserts that $P(x)$ is true for all $x$ and that $Q(y)$ is true for all $y$.
The second statement asserts that $P(x) wedge Q(x)$ is true for all $x$ and $y$, but this thus implies that $P(x)$ is true for all $x$ and $y$ and $Q(x)$ is true for all $x$ and $y$, which is the same thing.
answered Jan 30 at 19:53
Trevor KafkaTrevor Kafka
5309
answered Jan 30 at 19:53
Trevor KafkaTrevor Kafka
5309
answered Jan 30 at 19:53
Trevor KafkaTrevor Kafka
5309
answered Jan 30 at 19:53
Trevor KafkaTrevor Kafka
5309
5309
add a comment |
add a comment |
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2
$begingroup$
No difference...
$endgroup$
– Mauro ALLEGRANZA
Jan 30 at 19:42
2
$begingroup$
The first statement has 2 colons.
$endgroup$
– DanielV
Jan 30 at 20:59