Relations & Their Properties: Selected Exercises

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Exercise 10

Which relations in Exercise 4 are irreflexive?

A relation is irreflexive a A (a, a) R.

Ex. 4 relations on the set of all people:

a) a is taller than b.

b) a and b were born on the same day.

c) a has the same first name as b.

d) a and b have a common grandparent.

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Exercise 20

Must an asymmetric relation be antisymmetric?

A relation is asymmetric a b ( aRb (b, a) R ).

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Exercise 20Must an asymmetric relation be antisymmetric?

A relation is asymmetric a b ( aRb (b, a) R ).

To Prove:

(a b ( aRb (b, a) R ) ) (a b ( (aRb bRa ) a = b ) )

Proof:

1. Assume R is asymmetric.

2. a b ( ( a, b ) R ( b, a ) R ). (step 1. & defn of )

3. a b ( ( aRb bRa ) a = b ) (implication premise is false.)

4. Therefore, asymmetry implies antisymmetry.

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Exercise 20 continuedMust an antisymmetric relation be asymmetric?

(a b ( ( aRb bRa ) a = b ) ) a b ( aRb ( b, a ) R )?

Work on this question in pairs.

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Exercise 20 continued

Must an antisymmetric relation be asymmetric ?

(a b ( (aRb bRa ) a = b ) ) a b ( aRb (b, a) R ) ?

Proof that the implication is false:

1. Let R = { (a, a) }.

2. R is antisymmetric.

3. R is not asymmetric: aRa (a, a) R is false.

Antisymmetry thus does not imply asymmetry.

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Exercise 30

• Let R = { (1, 2), (1, 3), (2, 3), (2, 4), (3, 1) }.

• Let S = { (2, 1), (3, 1), (3, 2), (4, 2) }.

• What is S R?

1

2

34

R S1

2

34

S R

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Exercise 50

Let R be a relation on set A.

Show:

R is antisymmetric R R-1 { ( a, a ) | a A }.

To prove:

1. R is antisymmetric R R-1 { ( a, a ) | a A }

We prove this by contradiction.

2. R R-1 { ( a, a ) | a A } R is antisymmetric.

We prove this by contradiction.

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Exercise 50

Prove R is antisymmetric R R-1 { ( a, a ) | a A }.

1. Proceeding by contradiction, we assume that:

1. R is antisymmetric: a b ( ( aRb bRa ) a = b ).

2. It is not the case that R R-1 { ( a, a ) | a A }.

2. a b (a, b) R R-1, where a b. (Step 1.2)

3. Let (a, b) R R-1, where a b. (Step 2)

4. aRb , where a b. (Step 3)

5. aR-1b, where a b. (Step 3)

6. bRa, where a b. (Step 5 & defn of R-1)

7. R is not antisymmetric, contradicting step 1. (Steps 4 & 6)

8. Thus, R is antisymmetric R R-1 { ( a, a ) | a A }.

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Exercise 50 continued

Prove R R-1 { ( a, a ) | a A } R is antisymmetric.

1. Proceeding by contradiction, we assume that:

1. R R-1 { ( a, a ) | a A }.

2. R is not antisymmetric: ¬a b ( ( aRb bRa ) a = b )

2. Assume a b ( aRb bRa a b ) (Step 1.2)

3. bR-1a, where a b. (Step 2s & defn. of R-1)

4. ( b, a ) R R-1 where a b, contradicting step 1. (Step 2 & 3)

5. Therefore, R is antisymmetric.