Math Problem Statement

If A = {y, o, g, u, r, t} and U = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}, find A'.

Solution

To solve this problem, we need to find AA', which is the complement of set AA within the universal set UU. The complement of AA consists of all elements in UU that are not in AA.

Given:

  • A={y,o,g,u,r,t}A = \{ y, o, g, u, r, t \}
  • U={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}U = \{ a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z \}

Step 1: List the elements in AA

AA contains: y,o,g,u,r,ty, o, g, u, r, t.

Step 2: Identify elements in UU but not in AA

We go through each element in UU and exclude those found in AA:

So, the elements in UU but not in AA are: A={a,b,c,d,e,f,h,i,j,k,l,m,n,p,q,s,v,w,x,z}A' = \{ a, b, c, d, e, f, h, i, j, k, l, m, n, p, q, s, v, w, x, z \}

Final Answer

The complement AA' is: A={a,b,c,d,e,f,h,i,j,k,l,m,n,p,q,s,v,w,x,z}A' = \{ a, b, c, d, e, f, h, i, j, k, l, m, n, p, q, s, v, w, x, z \}

Would you like a breakdown of any specific steps?

Here are some additional questions you might find helpful:

  1. What is a universal set and how does it affect finding complements?
  2. How can we represent sets and their complements using Venn diagrams?
  3. How would we handle finding the complement if UU were infinite (like all integers)?
  4. What is the difference between a union and a complement?
  5. How would the answer change if another element were added to set AA?

Tip: When working with set complements, always double-check that you've excluded only the elements in the subset from the universal set.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Set Complement

Formulas

A' = U - A

Theorems

Complement of a Set

Suitable Grade Level

Grades 6-8