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# 1 Logic operations

For the following questions you will use X and Y defined as follows

X <- rep(c(TRUE, FALSE), 2)
X
[1]  TRUE FALSE  TRUE FALSE
Y <- rep(c(TRUE, FALSE), c(2, 2))
Y
[1]  TRUE  TRUE FALSE FALSE

These are all the possible combinations of two logic variables. Now we want to see what happens when we combine them

## 1.1 Evaluate “X and Y”

[1]  TRUE FALSE FALSE FALSE
# write here

## 1.2 Evaluate “X or Y”

[1]  TRUE  TRUE  TRUE FALSE
# write here

## 1.3 Evaluate “X and not Y”

[1] FALSE FALSE  TRUE FALSE
# write here

## 1.4 Evaluate “not X and not Y”

[1] FALSE FALSE FALSE  TRUE
# write here

## 1.5 Show that “not X and not Y” is the negation of “X or Y”. Print both results in different lines, so we can see they are the same. This is called De Morgan’s rule.

[1] FALSE FALSE FALSE  TRUE
[1] FALSE FALSE FALSE  TRUE
# write here

## 1.6 Show that “not X or not Y” is the negation of “X and Y”. Print both results in different lines, so we can see they are the same. This is also called De Morgan’s rule.

[1] FALSE  TRUE  TRUE  TRUE
[1] FALSE  TRUE  TRUE  TRUE
# write here

## 1.7 (Bonus) Can you prove that X and Y are really all combinations of two logic variables?

Write your comment here. Keep the > and delete the rest

# 2 Combination of three logic variables

Consider now A, B, and C, defined as follows

A <- rep(c(TRUE, FALSE), 4)
A
[1]  TRUE FALSE  TRUE FALSE  TRUE FALSE  TRUE FALSE
B <- rep(c(TRUE, TRUE, FALSE, FALSE), 2)
B
[1]  TRUE  TRUE FALSE FALSE  TRUE  TRUE FALSE FALSE
C <- rep(c(TRUE, FALSE), c(4, 4))
C
[1]  TRUE  TRUE  TRUE  TRUE FALSE FALSE FALSE FALSE

These are all the combinations of three logic values.

## 2.1 Show that “A and the result ofB or C” is equivalent to “the result of A and B, or the result of A and C”. Print both results in different lines, so we can see they are the same. This is called Distributive rule.

[1]  TRUE FALSE  TRUE FALSE  TRUE FALSE FALSE FALSE
[1]  TRUE FALSE  TRUE FALSE  TRUE FALSE FALSE FALSE
# write here

## 2.2 Show that “A or the result ofB and C” is equivalent to “the result of A or B, and the result of A or C”. Print both results in different lines, so we can see they are the same. This is also called Distributive rule.

[1]  TRUE  TRUE  TRUE FALSE  TRUE FALSE  TRUE FALSE
[1]  TRUE  TRUE  TRUE FALSE  TRUE FALSE  TRUE FALSE
# write here

## 2.3 (Bonus) Can you show the associative rule?

# write here

## 2.4 (Bonus) Can you show the commutative rule?

# write here

# 3 Modifying a vector

In contrast to single-value variables, when we use indices to modify a vector, it changes on place.

The answers to the following questions should work for any vector v. For the sake of example, consider the vector v defined as

v <- seq(from=7, to=1)
v
[1] 7 6 5 4 3 2 1
# write here

## 3.1 Change the first element to 8 and show the updated vector v.

[1] 8 6 5 4 3 2 1
# write here

## 3.2 Change the sign of the second element and show the updated vector v.

[1]  8 -6  5  4  3  2  1
# write here

## 3.3 Add 7 to the third and fourth element and show the updated vector v.

[1]  8 -6 12 11  3  2  1
# write here

## 3.4 Change again the sign of the second element and show the updated vector v.

[1]  8  6 12 11  3  2  1
# write here

## 3.5 Increase the last three elements by 10%.

[1]  8.0  6.0 12.0 11.0  3.3  2.2  1.1
# write here