Question

Which of the following statements are valid? (a) \(A \cup A=A\) (b) \(A \cap A=A\) (c) \(A \cup \varnothing=A\) \((d) A \cup U=U\) \((e) A \cap \varnothing=\varnothing\) \((f) A \cap U=A\) (g) The complement of \(\tilde{A}\) is \(A\)

Short answer

All statements (a) to (g) are valid.

Step-by-step solution

Identify Key Concepts

To determine the validity of the given statements, we need to apply concepts from set theory. Important concepts here include Union (\(\cup\)), Intersection (\(\cap\)), the Empty Set (\(\varnothing\)), and the Universal Set (\(U\)). We must also consider set complement properties.

Analyze Statement (a)

The statement \(A \cup A = A\) is valid. Union of a set with itself is the set itself, as no new elements are added.

Analyze Statement (b)

The statement \(A \cap A = A\) is valid. Intersection of a set with itself will produce the same set, because all elements common to \(A\) and \(A\) are those in \(A\).

Analyze Statement (c)

The statement \(A \cup \varnothing = A\) is valid. The union of a set with the empty set results in the original set, as \(\varnothing\) does not contribute any elements.

Analyze Statement (d)

The statement \(A \cup U = U\) is valid. The union of any set \(A\) with the universal set \(U\) results in \(U\), since \(U\) contains all possible elements.

Analyze Statement (e)

The statement \(A \cap \varnothing = \varnothing\) is valid. The intersection of any set \(A\) with the empty set is \(\varnothing\), because there are no common elements.

Analyze Statement (f)

The statement \(A \cap U = A\) is valid. The intersection of any set \(A\) with the universal set \(U\) results in \(A\), as all elements of \(A\) are also in \(U\).

Analyze Statement (g)

The statement relating to set complement states the complement of \(\tilde{A}\) is \(A\). This is valid, since by definition, the complement of the complement of \(A\) returns back to \(A\).

Key concepts

Union of Sets

In set theory, the union of two or more sets is a fundamental concept. Imagine turning on a flashlight in a dark room that has two separate pools of light. The union of these lights is essentially the larger area that the light covers when both pools overlap and combine. The union of sets is similar. It combines all the unique elements that are present in all sets.
For sets "A" and "B", the union is written as \(A \cup B\). It includes every element that is either in set "A", in set "B", or in both. Here, adding the same item twice doesn't change the union.

• For example, if \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\), then \(A \cup B = \{1, 2, 3, 4, 5\}\).
• Additionally, when you take the union of a set with itself, \(A \cup A\) is simply \(A\). Nothing new is added, so the set stays the same.
• If you then perform a union with the empty set, \(A \cup \varnothing\), it also remains \(A\) since the empty set doesn't add anything.
• Finally, the union of any set with the universal set \(U\) results in the universal set \(U\). All possible elements are already included.

Intersection of Sets

The intersection of sets is akin to drawing only the common parts that multiple sets share. Imagine two overlapping circles, where the overlap represents their intersection.
The intersection operation, noted as \(A \cap B\), includes only elements common to both set "A" and set "B". If an element is not shared by both, it's left out. This operation is helpful when you want to find similarities.

• For instance, if \(A = \{1, 2, 3\}\) and \(B = \{2, 3, 4\}\), then \(A \cap B = \{2, 3\}\). These are the elements that both sets share.
• When you look at the intersection of a set with itself, \(A \cap A\), you multiply not but only reaffirm that it's still \(A\), as every element is already part of itself.
• However, the intersection of any set with the empty set \(\varnothing\) results in an empty set. No elements can be shared because the empty set has nothing in it.
• Interestingly, when intersecting a set with the universal set \(U\), \(A \cap U\) just brings back set \(A\). All elements in \(A\) belong to the universal set by definition.

Empty Set

The empty set is like a box that holds nothing. It's important in mathematics because it's the starting or neutral point in many scenarios. When talking about sets, an empty set has zero elements. Think of it as a completely empty bag of cookies.

• It is denoted by \(\varnothing\) or sometimes written as \(\{\}\).
• The empty set is unique because it is the only set with no elements. A set without cookies is different from a set having a cookie with zero protein.
• When using the empty set in set operations, you see how it behaves as a form of identity:
  • Union Operation: \(A \cup \varnothing = A\) - You keep the contents of set \(A\), untouched.
  • Intersection Operation: \(A \cap \varnothing = \varnothing\) - Without anything in common, the result defaults to \(\varnothing\).
• Each time an empty set participates in these operations, it reinforces its invisibility, remaining a vital part of the set equation without changing the outcome.

Universal Set

Envision a grand library, encompassing every book imaginable. Similarly, the universal set, usually denoted by \(U\), captures all conceivable elements within a specific context or discussion. It is vast and complete.

• A universal set is defined by the context it serves. One might be all the natural numbers, another all the colors in a rainbow.
• In practical terms, working with a universal set helps define what's included in the whole universe of the relevant problem.
• When combining a set with this universal set:
  • Under union, \(A \cup U = U\). The universal set encompasses everything, setting everything into its grand collection.
  • On intersection, \(A \cap U = A\). Here, the focus shifts solely back to \(A\) because every member of \(A\) exists within the universal borders.
• The universal set provides a backdrop that contextualizes all set-related discussions.