d420 Discrete Math: Functions and Relations - Set 1 - Part 1

Correct Answer: ¬p

The negation of a statement p is represented as ¬p, which means "not p." This is the correct way to negate a logical statement.

Correct Answer: p∨q

A disjunction is represented by p∨q, which means "p or q." This is true if at least one of the statements is true.

Correct Answer: False

A logical implication p→q is false only when p is true and q is false.

Correct Answer: When p and q are both true or both false

A biconditional statement p↔q is true only when p and q have the same truth values.

Correct Answer: ¬q→¬p

The contrapositive of p→q is ¬q→¬p, which flips and negates both statements.

Correct Answer: ¬p∧¬q

De Morgan's Law states that the negation of p∨q is ¬p∧¬q.

Correct Answer: The statement is true under all possible interpretations

A tautology is a statement that is always true, regardless of the truth values of its components.

Correct Answer: A statement that is always false

A contradiction is a statement that is always false, regardless of the truth values of its components.

Correct Answer: A set containing no elements

The empty set is defined as the set with no elements.

Correct Answer: A set containing 8 elements

The power set of a set with n elements contains 2n subsets. For a set with 3 elements, the power set contains 23=8 subsets.

Correct Answer: A∪B

The union A∪B represents the set of all elements that are in A, in B, or in both.

Correct Answer: A∩B

The intersection A∩B is the set of elements that are in both A and B.

Correct Answer: A−B

The set difference A−B contains elements that are in A but not in B.

Correct Answer: The set of all ordered pairs (a,b) where a∈A and b∈B

The Cartesian product A×B is the set of all ordered pairs formed by taking elements from A and B.

Correct Answer: Every element in the set relates to itself

A reflexive relation requires that every element in the set is related to itself.

Correct Answer: If a is related to b and b is related to a, then a=b

In an antisymmetric relation, if two elements relate to each other in both directions, they must be equal.

Correct Answer: If a is related to b, then b is also related to a

A symmetric relation ensures that if one element is related to another, the second element is also related to the first.

Correct Answer: If a is related to b and b is related to c, then a is related to c

A transitive relation ensures that if an element relates to a second element, and the second element relates to a third, then the first element relates to the third.

Correct Answer: Reflexive, symmetric, and transitive

An equivalence relation must be reflexive, symmetric, and transitive.

Correct Answer: A graph with edges that have a direction

In a directed graph, edges are associated with a direction from one vertex to another.