Unit 6

Question 1

What is the cardinality of the set derived from the list of integers [4, 6, 8, 12, 6].

Answer 1

4

The cardinality of a set is the number of members within that set. In this case, we would ignore the second 6 when creating the set, leaving us with 4 members in total.

Question 2

Suppose that:

A is the set {john, mary, anil}

B is the set {anil, mary}

C is the set ∅.

Which of the following is true?

Select one or more:

• A U B = A
• A C B
• A n C = ∅
• C - B = {anil, mary}
• {anil, mary} ∊ A n B

Answer 2

• A U B = A
• A n C = ∅

Question 3

Using the attached tables, what would be the result of the following SQL query?

SELECT *

FROM student

Answer 3

Question 4

Using the attached tables, what would be the result of the following SQL query?

SELECT FirstName, LastName

FROM student

WHERE ModuleCode = ‘M269’

Answer 4

Question 5

Using the attached tables, what would be the result of the following SQL query?

SELECT FirstName, LastName, Level

FROM student CROSS JOIN module

WHERE Module = ModuleCode

Answer 5

Question 6

What is a set?

Answer 6

A set is an unordered collection of distinct entities.

For example, the list of numbers [4, 5, 8, 6, 4, 3, 4, 5, 8, 3, 7, 5, 7] would be formed into the following set (the order of the set is unimportant):

{3, 4, 5, 6, 7, 8}

Question 7

What is a set enumeration?

Answer 7

A set enumeration is when a set is written with all elements explicitly listed, for example, {3, 4, 5, 6, 7, 8}.

Question 8

What is a set comprehension?

Answer 8

A set comprehension is used to describe a set where the enumeration would be too complicated, or too lengthy to write down.

For example, the set comprehension {x : x is a whole number between 3 and 8} would describe the set enumeration {3, 4, 5, 6, 7, 8}.

Question 9

What would the set be for the following list of numbers:

5, 2, 8, 3, 5, 6, 2, 5, 9, 4, 6, 7

Answer 9

{2, 3, 4, 5, 6, 7, 8, 9}

Question 10

What would be the set enumeration described by the following set comprehension:

{x : x is a whole number between 23 and 34}

Answer 10

{23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}

Question 11

How would you write an empty set?

Answer 11

ø or {}.

Remember that the set of an empty set is not the same as the empty set itself, so the set {ø} is not the same as ø.

Question 12

What do the following expressions mean?

Answer 12

• x is a member of the set ‘S’, or x is an element of the set ‘S’
• x is not a member of the set ‘S’, or x is not an element of the set ‘S’

Question 13

What do the following expressions mean?

Answer 13

• The empty set is a subset of set ‘A’
• Set ‘A’ is a subset of itself
• Set ‘A’ is a proper subset of set ‘B’

Question 14

What does the following expression mean, when dealing with sets?

A = B

Answer 14

Set ‘A’ is equal to set ‘B’.

Equality of sets can also be defined in terms of subsets:

A = B if A is a proper subset of B, and B is a proper subset of A.

Question 15

What do the following expressions mean, and what would be the resulting sets?

Answer 15

• The union of sets {1, 2, 3} and {3, 4, 5}. The resulting set would be {1, 2, 3, 4, 5}
• The union of the empty set and set ‘A’. The resulting set would be set ‘A’

Question 16

What do the following expressions mean, and what would be the resulting sets?

Answer 16

• The intersection of sets {1, 2, 3} and {3, 4, 5}. The resulting set would be {3}.
• The intersection of the empty set and set ‘A’. The resulting set would be the empty set.

Question 17

What do the following expressions mean, and what would be the resulting sets?

Answer 17

All these expressions show the difference between two sets.

The resulting sets would be:

• {a, c, e}
• {1, 2}
• {5, 6}
• {x : x is an odd number}

Question 18

What is a tuple?

Answer 18

A tuple is a sequence of elements which is ordered and can contain duplicates. A tuple is written with round brackets rather than curly brackets.

Two tuples are only the same if they have the same elements in the same order.

E.g. (3, 1, 1) is not the same as (1, 1, 3) or (3, 1).

A 0-tuple with no members is written as ().

Question 19

What is the Cartesian product?

Answer 19

The Cartesian product for two sets is the set of all the ordered pairs (x, y), where x is a member of set ‘A’ and y is a member of set ‘B’.

Question 20

What is the Cartesian product of the following two sets?

{1, 2} and {1, 3, 5}

Answer 20

{(1, 1), (1, 3), (1, 5), (2, 1), (2, 3), (2, 5)}

Question 21

What is a well-formed formula (WFF) of propositional logic?

Answer 21

Given a set of propositional variables, {}, a well-formed formula (WFF) of propositional logic is any of:

• a propositional variable
• ¬A, where A is a WFF
• (A ∧ B), where A and B are WFFs
• (A ∨ B), where A and B are WFFs
• (A → B), where A and B are WFFs

Nothing else is a WFF.

Question 22

What is an interpretation of a set of WFFs?

Answer 22

An interpretation of a set of WFFs is an assignment of one of the truth values (TRUE or FALSE) to each of the variables.

Question 23

When is a WFF satisfiable?

Answer 23

A WFF is satisfiable only if there exists at least one interpretation in which that WFF is TRUE.

A set of WFFs is satisfiable only if there exists at least one interpretation in which every WFF in the set is TRUE.

In these cases, the interpretation is said to satisfy the WFF or WFFs.

Question 24

What is a contradiction?

Answer 24

A contradiction is a formula which is FALSE in every possible interpretation.

Question 25

What is a tautology?

Answer 25

A tautology is a formula which is TRUE in every possible interpretation.

Question 26

When is a formula contingent?

Answer 26

A formula is contingent if it is TRUE in some possible interpretations and FALSE in others.

Question 27

When would two WFFs be logically equivalent, and how is this written?

Answer 27

Given two WFFs, A₁ and A₂, A₁ is logically equivalent to A₂ if A₁ and A₂ have the same value in all interpretations.

This is written as A₁ ⇔ A₂.

Question 28

What is a valid argument?

Answer 28

An argument is valid only if there is no interpretation in which the premises are TRUE and the conclusion is FALSE.

Question 29

Is the following argument valid?

R ∧ S

¬R ∨ Q

S → Q

Answer 29

The argument is valid.

There is only one interpretation in which both premises are TRUE (the interpretation where Q, R and S are all TRUE). as the conclusion is also TRUE in this interpretation, the argument is valid.

Question 30

Is the following argument valid?

David is a programmer, or Sara is a doctor

If Sara is not a doctor, then Michael is a salesman

David is a programmer or Michael is a salesman

Answer 30

The argument is invalid.

Assuming we assign variables to each of the three basic propositions:

• D: David is a programmer
• S: Sara is a doctor
• M: Michael is a salesman

We can then translate the English argument to logic:

D ∨ S

¬S → M

D ∨ M

There is an interpretation in which the premises are TRUE, but the conclusion is FALSE. In that interpretation, D = FALSE, M = FALSE and S = TRUE. The premises are TRUE, but the conclusion is FALSE, therefore the argument is invalid.