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In each of the cases below, state whether or not the given relation R is reflexive, anti-reflexive, symmetric, anti-symmetric, transitive. Explain your five answers in each case. (a) R=A={(x,y)∈Z×Z ∣ x+2y= 1/3 }; (b) R=B={(x,y)∈Z×Z ∣ ∣x∣+∣y∣=4}; (c) R=C={(x,y)∈R×R ∣ xy≤1}; (d) R=D={(x,y)∈Q×Q ∣ ∣x+y∣>1};

User Paul Odeon
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Explanation:

Let's analyze each of the given relations R in terms of being reflexive, anti-reflexive, symmetric, anti-symmetric, and transitive:

(a) R=A={(x,y)∈Z×Z ∣ x+2y=1/3}:

- Reflexive: This relation is not reflexive because, for any integer x, the pair (x, x) does not satisfy the equation x + 2x = 1/3.

- Anti-reflexive: It is not anti-reflexive because there are pairs (x, x) that do not satisfy the given condition.

- Symmetric: This relation is symmetric because if (x, y) is in R, then (y, x) is also in R, as changing the order of x and y doesn't affect whether they satisfy the equation.

- Anti-symmetric: It is not anti-symmetric because there exist pairs (x, y) and (y, x) in R where x ≠ y.

- Transitive: This relation is not transitive because having (x, y) and (y, z) in R does not necessarily imply (x, z) is in R.

(b) R=B=(x,y)∈Z×Z ∣ :

- Reflexive: This relation is reflexive because for any integer x, the pair (x, x) satisfies the equation |x| + |x| = 4.

- Anti-reflexive: It is not anti-reflexive because it satisfies the reflexive property.

- Symmetric: This relation is symmetric because if (x, y) is in R, then (y, x) is also in R.

- Anti-symmetric: It is not anti-symmetric because there exist pairs (x, y) and (y, x) in R where x ≠ y.

- Transitive: This relation is not transitive because having (x, y) and (y, z) in R does not necessarily imply (x, z) is in R.

(c) R=C={(x,y)∈R×R ∣ xy≤1}:

- Reflexive: This relation is reflexive because for any real number x, the pair (x, x) satisfies the condition xy ≤ 1.

- Anti-reflexive: It is not anti-reflexive because it satisfies the reflexive property.

- Symmetric: This relation is symmetric because if (x, y) is in R, then (y, x) is also in R.

- Anti-symmetric: It is not anti-symmetric because there exist pairs (x, y) and (y, x) in R where x ≠ y.

- Transitive: This relation is transitive because if (x, y) and (y, z) are in R, then (x, z) is also in R due to the multiplication property of real numbers.

(d) R=D=>1:

- Reflexive: This relation is not reflexive because, for any rational number x, the pair (x, x) does not satisfy the condition |x + x| > 1.

- Anti-reflexive: This relation is anti-reflexive because it does not satisfy the reflexive property.

- Symmetric: This relation is not symmetric because there may be pairs (x, y) in R where (y, x) is not in R.

- Anti-symmetric: It is anti-symmetric because if (x, y) and (y, x) are in R, then x + y > 1 and y + x > 1, which implies x = y. Therefore, (x, y) = (y, x) only when x = y.

- Transitive: This relation is not transitive because having (x, y) and (y, z) in R does not necessarily imply (x, z) is in R.

User Matiferrigno
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