Binary Relations II (Lowersixth Science Mathematics)

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True/false on Cartesian products, Venn diagrams, and partial orders.

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Binary Relations II (Lowersixth Science Mathematics)


Binary Relations II (Lowersixth Science Mathematics)Version en ligne True/false on Cartesian products, Venn diagrams, and partial orders. par YAKILI LMS 1 A × B is commutative for all sets A and B. Yes No 2 The intersection of A and B always has more elements than the union of A and B. Yes No 3 The diagonal of A × A contains all pairs (a,a) with a in A. Yes No 4 A partial order can be visualized by Hasse diagrams. Yes No 5 In a Venn diagram with two sets, the overlap region represents elements common to both sets. Yes No 6 The universal set in a Venn diagram can be any finite set chosen for the problem. Yes No 7 The set A × B is always finite if A and B are finite. Yes No 8 A partial order cannot have cycles. Yes No 9 The empty set is not a subset of any set. Yes No 10 The projection map from A × B to A is always injective. Yes No 11 The Cartesian product A × B consists of all ordered pairs (a,b) with a in A and b in B. Yes No 12 The intersection of A and B is always equal to A if A is a subset of B. Yes No 13 A partial order requires symmetry to be valid. Yes No 14 The map that assigns to each pair (a,b) its first component is a surjective function from A × B to A. Yes No 15 The empty set is a valid function from the empty domain to any codomain. Yes No 16 The relation 'divides' on natural numbers is a partial order. Yes No 17 In a poset, every two elements have a greatest lower bound and least upper bound if it is a lattice. Yes No 18 Transitivity does not imply antisymmetry in general. Yes No 19 The union of A and B contains all elements that are in A or in B or in both. Yes No 20 The subset relation ⊆ is a partial order on the power set of a given universal set. Yes No 21 The set A × ∅ is the empty set. Yes No 22 The empty set is a subset of every set. Yes No 23 A poset must be reflexive on its elements. Yes No 24 In a Venn diagram, a universal set U contains all elements under consideration. Yes No 25 A total order is a partial order with complete comparability. Yes No 26 A Cartesian product is non-empty if both A and B are non-empty. Yes No 27 The intersection of two sets is a subset of both sets. Yes No 28 A Venn diagram with two circles can represent intersection, union, and difference regions. Yes No 29 If A = {1,2} and B = {3,4}, then A × B has 4 elements. Yes No 30 The order relation ≤ on real numbers is a partial order. Yes No 31 The cardinality of A × B equals the product of the cardinalities of A and B. Yes No 32 In a Venn diagram, the union region is only outside the circles. Yes No 33 The Cartesian product of two sets depends on the elements, not only their sizes. Yes No 34 The map that assigns to each pair (a,b) its second component is a surjective function from A × B to B. Yes No 35 For any sets A and B, \|A × B\| = \|A\| × \|B\|. Yes No 36 A Venn diagram with three sets can illustrate all possible region combinations. Yes No 37 A total order cannot be visualized with a Hasse diagram. Yes No 38 If A is empty, A × B is empty for any B. Yes No 39 For sets A and B, the projection maps from A × B to A and to B are well-defined. Yes No 40 The Cartesian product A × B has pairs where the first component comes from A and the second from B. Yes No 41 A partial order is a binary relation that is reflexive, antisymmetric, and transitive. Yes No 42 The power set of A is ordered by subset inclusion as a poset. Yes No 43 The Cartesian product is associative: (A × B) × C is isomorphic to A × (B × C). Yes No 44 In a Venn diagram, shading can indicate a complement region as well. Yes No 45 In a Venn diagram, the region inside a single circle corresponds to that set. Yes No 46 If A and B are non-empty, A ∩ B is always non-empty. Yes No 47 A universal relation on a set is the set of all ordered pairs (x,y) with x,y in the set. Yes No 48 In a Hasse diagram, edges indicate cover relations in a poset. Yes No 49 The complement of a region in a Venn diagram is represented by shading outside the region with respect to the universal set. Yes No 50 The subset relation ⊆ is antisymmetric on sets: if A ⊆ B and B ⊆ A then A = B. Yes No

Binary Relations II (Lowersixth Science Mathematics)

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Binary Relations II (Lowersixth Science Mathematics)Version en ligne

par YAKILI LMS

A × B is commutative for all sets A and B.

The intersection of A and B always has more elements than the union of A and B.

The diagonal of A × A contains all pairs (a,a) with a in A.

A partial order can be visualized by Hasse diagrams.

In a Venn diagram with two sets, the overlap region represents elements common to both sets.

The universal set in a Venn diagram can be any finite set chosen for the problem.

The set A × B is always finite if A and B are finite.

A partial order cannot have cycles.

The empty set is not a subset of any set.

The projection map from A × B to A is always injective.

The Cartesian product A × B consists of all ordered pairs (a,b) with a in A and b in B.

The intersection of A and B is always equal to A if A is a subset of B.

A partial order requires symmetry to be valid.

The map that assigns to each pair (a,b) its first component is a surjective function from A × B to A.

The empty set is a valid function from the empty domain to any codomain.

The relation 'divides' on natural numbers is a partial order.

In a poset, every two elements have a greatest lower bound and least upper bound if it is a lattice.

Transitivity does not imply antisymmetry in general.

The union of A and B contains all elements that are in A or in B or in both.

The subset relation ⊆ is a partial order on the power set of a given universal set.

The set A × ∅ is the empty set.

The empty set is a subset of every set.

A poset must be reflexive on its elements.

In a Venn diagram, a universal set U contains all elements under consideration.

A total order is a partial order with complete comparability.

A Cartesian product is non-empty if both A and B are non-empty.

The intersection of two sets is a subset of both sets.

A Venn diagram with two circles can represent intersection, union, and difference regions.

If A = {1,2} and B = {3,4}, then A × B has 4 elements.

The order relation ≤ on real numbers is a partial order.

The cardinality of A × B equals the product of the cardinalities of A and B.

In a Venn diagram, the union region is only outside the circles.

The Cartesian product of two sets depends on the elements, not only their sizes.

The map that assigns to each pair (a,b) its second component is a surjective function from A × B to B.

For any sets A and B, |A × B| = |A| × |B|.

A Venn diagram with three sets can illustrate all possible region combinations.

A total order cannot be visualized with a Hasse diagram.

If A is empty, A × B is empty for any B.

For sets A and B, the projection maps from A × B to A and to B are well-defined.

The Cartesian product A × B has pairs where the first component comes from A and the second from B.

A partial order is a binary relation that is reflexive, antisymmetric, and transitive.

The power set of A is ordered by subset inclusion as a poset.

The Cartesian product is associative: (A × B) × C is isomorphic to A × (B × C).

In a Venn diagram, shading can indicate a complement region as well.

In a Venn diagram, the region inside a single circle corresponds to that set.

If A and B are non-empty, A ∩ B is always non-empty.

A universal relation on a set is the set of all ordered pairs (x,y) with x,y in the set.

In a Hasse diagram, edges indicate cover relations in a poset.

The complement of a region in a Venn diagram is represented by shading outside the region with respect to the universal set.

The subset relation ⊆ is antisymmetric on sets: if A ⊆ B and B ⊆ A then A = B.