Last Updated on September 10, 2026 by Maged kamel
This post is about the Universal set and its subsets. Solved problems are introduced to clarify the idea.
Introduction to the Universal set and subsets of sets.
What is a subset?
I quote: If every element of A is also an element of B. If whenever x ∈A, as x is a part of A, then x ∈ B, then x is an element of B.
The graph shows A as a small circle inside the big circle B. This means that all the elements in A also exist in B.
This means A is a subset of B, which is logical. The shapes used are called Venn diagrams, a diagram style that shows the logical relation between sets. The symbol used is A⊆B. A is a subset of B. Set A is included in set B.
Suppose A intersects with B. Then A is partially in B but not fully. Then A is not a subset of B. To explain that, another symbol is used, which is A⊄B. Set A is not a subset of set B.
A solved problem 5.
Solved problem #5: If set A = {1,2,3,4,5,6}, set B {2,4,5}, and set C = {1,2,3,4,5}, it is required to show the Venn diagram.
We can represent each set as a circle. Set A includes numbers from 1 to 6; the second circle is for B, which contains 2, 4, and 5. At the same Time, C is another circle that includes 1, 2, 3, 4, and 5.
The biggest circle A is drawn first the numbers are from 1 to 6. We can select the part that includes 2, 4, and 5 and make other shapes. We can choose the C part, which includes 1, 2, 3, 4, and 5, as a new shape.
The only remaining part for A is number 6. A is the biggest shape, B is the smallest figure, and C is the middle shape. If we need to express ourselves, we start with the smallest and work up to the biggest.
Set B is a subset of A, and C is also a subset of A. Set B is a subset of C. If we start with the 5 largest shapes and move to the smallest, the largest shape is not a subset of B; A is not a subset of C; and C, which is bigger than B, is not a subset of B.
A solved problem #66.
A solved problem number #6, if set A={1,2,3,4}, if set B={1,4}, while set C={1}. Describe which is the subset. A is the biggest, followed by B, and C is the smallest. Also, check from smallest to biggest.
B is a subset of A for the smallest-to-biggest relation, since B is smaller than A. B contains 1,4, which are included in A. C is also a subset of A since element 1 is included in set A. C is a subset of B since element 1 is included in set B. But A is not a subset of B; A is > B. A is not a subset of C; A is >C.
Solved Problems-7-8.
Let us check the solved problem #7: set M = {a, b}. How many subsets are there? What are the different alternatives? We have φ, Ø. M has 4 substets,A={a}, B={b}, {a,b} ,Ø.
Solved problem 8:How many subsets of {a, a,b,c}?
The answer is that the subset A is {a}. B = {b}, C = {c}; later, start using the mix {a,b,c}, null. So far, we have selected 5 choices. Add the alternatives {a,c},{a,b},{b,c}. We have a total of 8 subsets, all are {a},{b},{c},{a},Ø = {},{a,b,c},{a,b},{a,c}.
Universal set.
The new item, Capital U, is drawn as a rectangle for a particular problem and is a set containing all the possible elements of that problem.
A solved problem 9.
Let us have a Solved problem-9, if U={1,2,3,4,5,6,7,8,9}, while A={1,2,5,6}. While B={5,6}.
Draw a Venn diagram to represent these sets. If we draw each set separately. This is the circle that represents A={1,2,5,6}
B={3,9}. This is U {1,2,3,4,5,6,7,8,9}. For the Venn diagram, draw a big box to represent U and draw shapes for A and B that include each element. For A, take {1,2,5,6} and draw a shape, then select B ={3,9} and draw a shape.
How many elements do we have in total? We have nine elements. We have already used four elements for A and two for B, so we expect to use the remaining three elements.
The remaining elements are 4, 7, and 8.
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For an external link, math is fun with a Venn diagram.
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