Mathematical Proofs (Relations and Functions)

For the sets A = {a, b, c} and B = {r,s,t, u}, let R = {(a,s), (a,t), (b,t)} be a relation from A to B. Determine dom(R)

Answer explanation

The domain of a relation is the set of all elements from the first set in the ordered pairs of the relation. In this case, the first set is A = {a, b, c}. However, there are no ordered pairs in R that have c as the first element. Therefore, the domain of R is {a, b}.

For the sets A = {a, b, c} and B = {r,s,t, u}, let R = {(a,s), (a,t), (b,t)} be a relation from A to B. Determine range(R)

Answer explanation

The range of a relation is the set of all second elements in the ordered pairs. In this case, the range of R is {s, t}.

For the relation R = {(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)} defined on the set {1, 2, 3}, what is R−1?

Answer explanation

The relation R-1 is the inverse of R, which means that the pairs in R are reversed. Therefore, R-1 = {(1, 1), (2, 1), (2, 2), (3, 1), (3, 2), (3, 3)}. This is the correct choice.

Determine the inverse relation R−1 for the relation R = {(x, y) : x + 4y is odd} defined on N.

Answer explanation

The inverse relation R−1 for R = {(x, y) : x + 4y is odd} is {(x, y) : y + 4x is odd}. This is the correct choice because it reverses the variables and the coefficients in the equation.

Let A and B be sets with |A|=|B| = 4. Prove or disprove: If R is a relation from A to B where |R| = 9 and R = R^−1, then A = B.

Answer explanation

The statement is false. If R = R^−1, then R is a symmetric relation. However, this does not imply that A = B. Counterexamples can be found where A and B are not equal.

Let A = {a, b, c, d} and let R = {(a, a), (a, b), (a, c), (a, d), (b, b), (b, c), (b, d), (c, c), (c, d), (d, d)} be a relation on A. Which of the properties reflexive, symmetric and transitive does the relation R possess?

• Reflexive

• Symmetric

• Transitive

Answer explanation

The relation R is reflexive because every element in A is related to itself. It is also transitive because if (a, b) and (b, c) are in R, then (a, c) is also in R.

Let S = {a, b, c}. Then R = {(a, b)} is a relation on S. Which of the properties reflexive, symmetric and transitive does the relation R possess?

• Reflexive

• Symmetric

• Transitive

Answer explanation

The relation R is transitive because it satisfies the property of transitivity, which means that if (a, b) and (b, c) are in R, then (a, c) must also be in R.

A relation R is defined on Z by aRb if |a − b| ≤ 2. Which of the properties reflexive, symmetric and transitive does the relation R possess?

• Reflexive

• Symmetric

• Transitive

Answer explanation

The relation R is reflexive because for every element a in Z, aRa is true. However, it is not symmetric or transitive.

Let R = ∅ be the empty relation on a nonempty set A. Which of the properties reflexive, symmetric and transitive does R possess?

• Reflexive

• Symmetric

• Transitive

Answer explanation

The empty relation R on a nonempty set A is symmetric and transitive, but not reflexive.

A relation R is defined on Z by xRy if x · y ≥ 0.

• Reflexive

• Symmetric

• Transitive

Answer explanation

The relation R is reflexive because for any integer x, x · x is always greater than or equal to 0.