Math Problem Statement

Given that Set P = {1, 6, 8, 22, 42}, Q = {52, a, b, c, d, e, f} and R = {1, 18, 42, 16, c, g, d, h}, determine Q ∪ (P - R).

Solution

To solve this problem, we need to calculate Q(PR)Q \cup (P - R) given the following sets:

  • P={1,6,8,22,42}P = \{1, 6, 8, 22, 42\}
  • Q={52,a,b,c,d,e,f}Q = \{52, a, b, c, d, e, f\}
  • R={1,18,42,16,c,g,d,h}R = \{1, 18, 42, 16, c, g, d, h\}

Step 1: Calculate PRP - R

The operation PRP - R represents the elements in PP that are not in RR.

  • Elements of PP: {1,6,8,22,42}\{1, 6, 8, 22, 42\}
  • Elements of RR: {1,18,42,16,c,g,d,h}\{1, 18, 42, 16, c, g, d, h\}

Comparing each element in PP with RR:

  • 11 is in RR, so it is excluded.
  • 66 is not in RR, so it is included.
  • 88 is not in RR, so it is included.
  • 2222 is not in RR, so it is included.
  • 4242 is in RR, so it is excluded.

Thus, PR={6,8,22}P - R = \{6, 8, 22\}.

Step 2: Calculate Q(PR)Q \cup (P - R)

Now, we find the union of QQ and PRP - R, denoted by Q(PR)Q \cup (P - R).

  • Elements of QQ: {52,a,b,c,d,e,f}\{52, a, b, c, d, e, f\}
  • Elements of PRP - R: {6,8,22}\{6, 8, 22\}

Taking the union of these two sets gives: Q(PR)={52,a,b,c,d,e,f,6,8,22}Q \cup (P - R) = \{52, a, b, c, d, e, f, 6, 8, 22\}

Final Answer:

Q(PR)={52,a,b,c,d,e,f,6,8,22}Q \cup (P - R) = \{52, a, b, c, d, e, f, 6, 8, 22\}

Would you like further clarification on any step?


Here are five related questions that may deepen your understanding:

  1. How do we determine the elements in PRP - R when both sets contain different types of elements (numbers and letters)?
  2. What is the meaning of the union operation in set theory?
  3. How would the result change if QQ or RR were different?
  4. What would happen if we calculated PRP \cap R instead of PRP - R?
  5. Can we determine QRQ \cap R with the given sets, and what would it represent?

Tip: Always double-check which elements are being removed in set difference operations, as it can easily lead to mistakes in complex sets.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Union
Set Difference

Formulas

A - B = {x | x ∈ A and x ∉ B}
A ∪ B = {x | x ∈ A or x ∈ B}

Theorems

Union of Sets
Set Difference

Suitable Grade Level

Grades 9-12