Question

The relation \(\leq\) on numbers has the following properties. (a) \(a \leq a \forall a \in R\) (Reflexivity) (b) If \(\mathrm{a} \leq \mathrm{b}\) and \(\mathrm{b} \leq \mathrm{a}\) then \(\mathrm{a}=\mathrm{b} \forall \mathrm{a}, \mathrm{b} \in \mathrm{R}\) (Antisymmetry) (c) If \(\mathrm{a} \leq \mathrm{b}\) and \(\mathrm{b} \leq \mathrm{c}\) then \(\mathrm{a} \leq \mathrm{c} \forall \mathrm{a}, \mathrm{b} \in \mathrm{R}\) (Transitivity) Which of the above properties the relation \(\subset\) on \(\mathrm{P}(\mathrm{A})\) has? (a) (i) and (ii) (b) (i) and (iii) (c) (ii) and (iii) (d) (i), (ii) and (iii)

Short answer

The relation ⊆ on P(A) has properties (a) - Reflexivity, (b) - Antisymmetry, and (c) - Transitivity. Hence, the correct answer is (d) (i), (ii) and (iii).

Step-by-step solution

Definition of Subset Relation and Power Set

For this exercise, the subset relation, denoted by ⊆, is a binary relation between sets. Specifically, set A is a subset of set B, written as A ⊆ B, if every element of A is also an element of B. The power set of A, denoted by P(A), is the set of all possible subsets of A, including the empty set and A itself. We will now examine properties (a), (b), and (c) for the relation ⊆ on P(A).

Property (a) - Reflexivity

Reflexive property states that a relation satisfies reflexivity if it is true for all elements in the domain of the relation. In other words, for the ⊆ relation on P(A), we must check if the statement \(A \subseteq A\) holds for all subsets A in P(A). Since every element in A is found in A, the statement \(A \subseteq A\) is always true. Therefore, the ⊆ relation is reflexive on P(A).

Property (b) - Antisymmetry

Antisymmetric property states that a relation R is antisymmetric if, for all elements a and b, whenever both R(a, b) and R(b, a) are true, then a = b must be true. For the ⊆ relation on P(A), we need to check if \(A \subseteq B\) and \(B \subseteq A\) implies \(A = B\). Since both conditions state that every element in A is in B, and every element in B is in A, it means that both sets contain the same elements and are equal. Therefore, the ⊆ relation is antisymmetric on P(A).

Property (c) - Transitivity

Transitive property states that a relation R is transitive if, for all elements a, b, and c, when R(a, b) and R(b, c) are true, then R(a, c) must also be true. For the ⊆ relation on P(A), we need to check if \(A \subseteq B\) and \(B \subseteq C\) implies \(A \subseteq C\). If every element in A is in B and every element in B is in C, then it follows that every element in A is also in C. Therefore, the ⊆ relation is transitive on P(A). In conclusion, the relation ⊆ on P(A) has properties (a) - Reflexivity, (b) - Antisymmetry, and (c) - Transitivity. Hence, the correct answer is: (d) (i), (ii) and (iii)

Key concepts

Reflexivity in Mathematics

Reflexivity is a foundational concept in mathematics, particularly in set theory and relations. An easy way to understand reflexivity is by considering it as a form of self-inclusion. For a relation to be reflexive within a set, each element must be related to itself. In simpler terms, for any element you pick from the set, it should have a relationship with itself according to the definition of the relation in question.

In the context of our exercise with the subset relation \(\subseteq\) on sets, this means that any set is considered to be a subset of itself. It's a bit like every person being a member of their 'self' family. This property is universally true for subsets because a set will always contain all the elements it has - which satisfies the very definition of reflexivity. Hence, given any set \(A\) from the power set \(P(A)\), the reflexive property holds because \(A \subseteq A\) is always true. This is a critical notion, as it provides consistency and a basis for more complex relations in mathematics.

Antisymmetry in Relations

While symmetry in everyday life often implies balance and equivalence, antisymmetry in mathematics has a unique application in set relations. Antisymmetry deals with the scenario where two elements are in relation to each other in both directions but are required to be identical for the relation to hold true. It may sound a little complex, but it’s actually quite a straightforward concept.

Let's take our subset example to clarify this: if \(A \subseteq B\) and \(B \subseteq A\), antisymmetry means that \(A\) and \(B\) must actually be the very same set. This is analogous to saying if person A is the sibling of person B and person B is the sibling of person A, they are siblings – but in the realm of sets, it means they are identical. It's a relationship that validates the equality between elements or sets when they are related in both ways. In essence, antisymmetry avoids redundancy, ensuring that a two-way relationship does not result in different sets but rather affirms their sameness.

Transitivity in Set Theory

Transitivity is a term that often conjures images of trains passing through stations, connecting one point to another seamlessly. In set theory, transitivity is the mathematical version of this concept — a relation is called transitive if, whenever one element is related to a second, and the second is related to a third, then the first must be related to the third as well.

For subsets, this is like a chain where each link securely connects to the next. If \(A \subseteq B\) and \(B \subseteq C\), transitivity ensures that \(A \subseteq C\). It bolsters the integrity of the relational structure, guaranteeing that the connection is maintained throughout a sequence of elements or sets. This property is especially important in the world of mathematics because it helps maintain logical consistency and supports the construction of proofs and theorems. Transitivity empowers mathematicians and students alike to make broader inferences from known relationships, reinforcing our understanding of mathematical structures.