2. Let P be the proposition \"Ann went to the park.\" Let Q be the proposition \

Question

2. Let P be the proposition "Ann went to the park." Let Q be the proposition "Ann got ice cream.” Let R be the proposition "Ann's family went to the park on Saturday." Let S be the proposition "Ann's family played tennis. Rewrite the following English statements in symbolic language using the variables P, Q, R, and S and the logical connectives: a) Ann's family did not play tennis and Ann did not get ice cream. b) Ann did not get ice cream but did go to the park. c) Ann got ice cream only if her family went to the park on Saturday. d) Only if her family did not go to the park on Saturday did Ann get ice cream. e) Either Ann went to the park, or Ann's family did not play tennis and did not go to the park on Saturday tIf Ann's family played tennis and Ann's family did not go to the park on Saturday then Ann did not go to the park

Explanation / Answer

Given the prepositions are.

P = Ann went to the park.

Q = Ann got ice cream.

R = Ann's family went to the park on Saturday.

S = Ann's family played tennis.

a) Ann's family did not play tennis and Ann did not get ice cream

~S = Ann's family did not play tennis

^ = and

~Q = Ann did not get ice cream.

Hence, ~S ^ ~Q is the required symbolic statement.

b) Ann did not get ice cream but did not go to the park.

~S = Ann did not get ice cream

^ ~P = but did not go to the park.

Hence, ~S ^ ~P is the required symbolic statement.

c) Ann got ice cream only if her family went to the park on Saturday.

Q = Ann got ice cream

R = her family went to the park on Saturday.

Since both the statements related positively to each other, hence we will use <=> against only if.

Q <=> R is the required symbolic statement.

d) Only if her family did not go the park on Saturday did Ann get ice cream.

In the above statement, one statement is negative and the other is positive and both are dependent on each other hence we will use =>

~R = her family did not go the park on Saturday

Q = Ann get ice cream.

Hence, ~R => Q is the required symbolic statement.

e) Either Ann went to the park, or Ann's family did not play tennis and did not go to the park on Saturday.

There are two individual statements in the above complete sentence.

1. Either Ann went to the park

or

2. Ann's family did not play tennis and did not go to the park on Saturday.

In the above statement or is used hence we will use the symbol v to OR both the statement and another symbol ^ for and-ing is used in the 2nd statement.

P = Ann went to the park.

~S = Ann's family did not play tennis.

~R = did not go to the park on Saturday.

(~S ^ ~R) = Ann's family did not play tennis and did not go to the park on Saturday.

P v (~S ^ ~R) = Either Ann went to the park, or Ann's family did not play tennis and did not go to the park on Saturday.

Hence, P v (~S ^ ~R) is the required symbolic statement.

f) If Ann's family played tennis and Ann's family did not go to the park on Saturday then Ann did not go to the park.

There are two statements contained in above sentence.

1. If Ann's family played tennis and Ann's family did not go to the park on Saturday

then

2. Ann did not go to the park.

Hence the symbols for first part is S ^ ~R

And since one of the statement is negative and other is positive hence we use =>

S ^ ~R => ~P, is the required symbolic statement.

a) Ann's family did not play tennis and Ann did not get ice cream

~S = Ann's family did not play tennis

^ = and

~Q = Ann did not get ice cream.

Hence, ~S ^ ~Q is the required symbolic statement.

b) Ann did not get ice cream but did not go to the park.

~S = Ann did not get ice cream

^ ~P = but did not go to the park.

Hence, ~S ^ ~P is the required symbolic statement.

c) Ann got ice cream only if her family went to the park on Saturday.

Q = Ann got ice cream

R = her family went to the park on Saturday.

Since both the statements related positively to each other, hence we will use <=> against only if.

Q <=> R is the required symbolic statement.

d) Only if her family did not go the park on Saturday did Ann get ice cream.

In the above statement, one statement is negative and the other is positive and both are dependent on each other hence we will use =>

~R = her family did not go the park on Saturday

Q = Ann get ice cream.

Hence, ~R => Q is the required symbolic statement.

e) Either Ann went to the park, or Ann's family did not play tennis and did not go to the park on Saturday.

There are two individual statements in the above complete sentence.

1. Either Ann went to the park

or

2. Ann's family did not play tennis and did not go to the park on Saturday.

In the above statement or is used hence we will use the symbol v to OR both the statement and another symbol ^ for and-ing is used in the 2nd statement.

P = Ann went to the park.

~S = Ann's family did not play tennis.

~R = did not go to the park on Saturday.

(~S ^ ~R) = Ann's family did not play tennis and did not go to the park on Saturday.

P v (~S ^ ~R) = Either Ann went to the park, or Ann's family did not play tennis and did not go to the park on Saturday.

Hence, P v (~S ^ ~R) is the required symbolic statement.

f) If Ann's family played tennis and Ann's family did not go to the park on Saturday then Ann did not go to the park.

There are two statements contained in above sentence.

1. If Ann's family played tennis and Ann's family did not go to the park on Saturday

then

2. Ann did not go to the park.

Hence the symbols for first part is S ^ ~R

And since one of the statement is negative and other is positive hence we use =>

S ^ ~R => ~P, is the required symbolic statement.

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