Binary Relations (Lowersixth Science Mathematics)

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Quick true/false on relations and equivalence

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Binary Relations (Lowersixth Science Mathematics)


Binary Relations (Lowersixth Science Mathematics)Version en ligne Quick true/false on relations and equivalence par YAKILI LMS 1 The set of all ordered pairs (a,b) ∈ A×A is always a function. Yes No 2 Antisymmetry means (a,b) ∈ R and (b,a) ∈ R never occur together. Yes No 3 If a relation is reflexive, it automatically has to be antisymmetric. Yes No 4 An equivalence relation is reflexive, symmetric, and transitive. Yes No 5 Transitivity implies symmetry automatically. Yes No 6 The diagonal {(a,a) \| a ∈ A} is always closed under any relation. Yes No 7 If (a,b) ∈ R then b must be equal to a for all a. Yes No 8 An ordered pair is written as (a,b) and is equal to (a,b) only if a=a and b=b. Yes No 9 A relation is called symmetric if it relates each pair in exactly one direction. Yes No 10 If a relation is transitive, then it cannot be symmetric. Yes No 11 The relation (a,b) ∈ R is the same as (b,a) ∈ R for all a,b. Yes No 12 For real numbers, the relation ≤ is a partial order on R, hence reflexive, antisymmetric, and transitive. Yes No 13 The concept of a relation is only defined for numbers. Yes No 14 Every relation on A is an equivalence relation. Yes No 15 A relation cannot be empty. Yes No 16 Antisymmetry property: if (a,b) ∈ R and (b,a) ∈ R, then a = b. Yes No 17 Reflexivity: for every a in A, (a,a) ∈ R. Yes No 18 The set of all elements equivalent to a given a under an equivalence relation partitions A. Yes No 19 If R is symmetric and reflexive, it must be transitive. Yes No 20 Symmetry property: if (a,b) ∈ R then (b,a) ∈ R. Yes No 21 The empty set is a non-relational set. Yes No 22 An irreflexive relation can never be antisymmetric. Yes No 23 Equality of ordered pairs: (a,b) = (c,d) iff a=c and b=d. Yes No 24 Every ordered pair is related to itself in every relation. Yes No 25 Transitivity: if (a,b) ∈ R and (b,c) ∈ R then (a,c) ∈ R. Yes No 26 For a function f, the graph is a subset of A×B with exactly one image per input, but this is unrelated to binary relations. Yes No 27 A relation is called total if it contains all possible ordered pairs. Yes No 28 A binary relation must always be reflexive to be meaningful. Yes No 29 A relation is a function if it relates every element to two or more images. Yes No 30 If R is an equivalence relation, it groups elements into disjoint equivalence classes. Yes No 31 A relation can be both symmetric and antisymmetric without being reflexive. Yes No 32 An antisymmetric relation is always reflexive. Yes No 33 Every equivalence relation on R has exactly one equivalence class. Yes No 34 If R is reflexive and transitive, it must be symmetric. Yes No 35 A symmetric relation cannot have any distinct related elements. Yes No 36 An equivalence relation cannot partition a set. Yes No 37 A binary relation on a set A is a subset of A × A. Yes No 38 The relation ‘is less than’ on integers is symmetric. Yes No 39 If a relation is antisymmetric, it cannot be symmetric. Yes No 40 The inverse of a relation R is always equal to R itself. Yes No 41 A binary relation on a set A is a subset of A × A. Yes No 42 A binary relation is always a function. Yes No 43 For a relation to be transitive, if (a,b) and (b,c) are in R, then (a,c) is in R. Yes No 44 If (a,b) in R then (b,a) in R automatically makes R antisymmetric. Yes No 45 All relations are transitive. Yes No 46 An equivalence relation on a set partitions the set into disjoint equivalence classes. Yes No 47 In an ordered pair (a, b), the first element is a and the second is b. Yes No 48 If (a,b) and (b,a) are in R, then a must equal b for all a and b. Yes No 49 If a relation R on a set is symmetric, then whenever (a,b) is in R, (b,a) is in R. Yes No 50 A reflexive relation must relate every element to every other element. Yes No

Binary Relations (Lowersixth Science Mathematics)

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Binary Relations (Lowersixth Science Mathematics)Version en ligne

par YAKILI LMS

The set of all ordered pairs (a,b) ∈ A×A is always a function.

Antisymmetry means (a,b) ∈ R and (b,a) ∈ R never occur together.

If a relation is reflexive, it automatically has to be antisymmetric.

An equivalence relation is reflexive, symmetric, and transitive.

Transitivity implies symmetry automatically.

The diagonal {(a,a) | a ∈ A} is always closed under any relation.

If (a,b) ∈ R then b must be equal to a for all a.

An ordered pair is written as (a,b) and is equal to (a,b) only if a=a and b=b.

A relation is called symmetric if it relates each pair in exactly one direction.

If a relation is transitive, then it cannot be symmetric.

The relation (a,b) ∈ R is the same as (b,a) ∈ R for all a,b.

For real numbers, the relation ≤ is a partial order on R, hence reflexive, antisymmetric, and transitive.

The concept of a relation is only defined for numbers.

Every relation on A is an equivalence relation.

A relation cannot be empty.

Antisymmetry property: if (a,b) ∈ R and (b,a) ∈ R, then a = b.

Reflexivity: for every a in A, (a,a) ∈ R.

The set of all elements equivalent to a given a under an equivalence relation partitions A.

If R is symmetric and reflexive, it must be transitive.

Symmetry property: if (a,b) ∈ R then (b,a) ∈ R.

The empty set is a non-relational set.

An irreflexive relation can never be antisymmetric.

Equality of ordered pairs: (a,b) = (c,d) iff a=c and b=d.

Every ordered pair is related to itself in every relation.

Transitivity: if (a,b) ∈ R and (b,c) ∈ R then (a,c) ∈ R.

For a function f, the graph is a subset of A×B with exactly one image per input, but this is unrelated to binary relations.

A relation is called total if it contains all possible ordered pairs.

A binary relation must always be reflexive to be meaningful.

A relation is a function if it relates every element to two or more images.

If R is an equivalence relation, it groups elements into disjoint equivalence classes.

A relation can be both symmetric and antisymmetric without being reflexive.

An antisymmetric relation is always reflexive.

Every equivalence relation on R has exactly one equivalence class.

If R is reflexive and transitive, it must be symmetric.

A symmetric relation cannot have any distinct related elements.

An equivalence relation cannot partition a set.

A binary relation on a set A is a subset of A × A.

The relation ‘is less than’ on integers is symmetric.

If a relation is antisymmetric, it cannot be symmetric.

The inverse of a relation R is always equal to R itself.

A binary relation on a set A is a subset of A × A.

A binary relation is always a function.

For a relation to be transitive, if (a,b) and (b,c) are in R, then (a,c) is in R.

If (a,b) in R then (b,a) in R automatically makes R antisymmetric.

All relations are transitive.

An equivalence relation on a set partitions the set into disjoint equivalence classes.

In an ordered pair (a, b), the first element is a and the second is b.

If (a,b) and (b,a) are in R, then a must equal b for all a and b.

If a relation R on a set is symmetric, then whenever (a,b) is in R, (b,a) is in R.

A reflexive relation must relate every element to every other element.