Introduction to Proofs - Functions, Relations, & Cardinality

$\A= \left{\varnothing, 3, 11, 13, 16, 19\right} \B= \left{\varnothing, 3, 4, 11, 16\right} \C= \left{\varnothing, 2, 12, 19\right}$

Find the cardinality of $A\cup A$ .

Explanation

The question is asking us to find the number of elements in the union of A and A .

We simply combine the two sets together, and count the number of elements. If there are any elements that are the same, we keep one of them, and not both.

$A\cup A = \left{\varnothing, 3, 11, 13, 16, 19\right}$

Now we simply count the number of elements in this set.

$\A= \left{3, 4, 9, 10, 11, 16\right} \B= \left{2, 7, 10, 15, 16\right} \C= \left{\varnothing, 8, 14, 17, 18\right}$

Find the cardinality of $A\cup A$ .

Explanation

The question is asking us to find the number of elements in the union of A and A .

We simply combine the two sets together, and count the number of elements. If there are any elements that are the same, we keep one of them, and not both.

$A\cup A = \left{3, 4, 9, 10, 11, 16\right}$

Now we simply count the number of elements in this set.

$\A= \left{4, 12, 20, 25, 28\right} \B= \left{\varnothing, 4, 8, 19, 22, 28\right} \C= \left{\varnothing, 3, 4, 8, 13, 14\right}$

Find the cardinality of $B\cap B$

Explanation

The question is asking us to find the number of elements in the intersection of B and B .

We simply find the common elements of the two sets, and count the number of elements.

$B\cap B = \left{\varnothing, 4, 8, 19, 22, 28\right}$
Now we simply count the number of elements in this set.

$\A= \left{9, 10, 16, 17, 19\right} \B= \left{3, 6, 8, 18\right} \C= \left{1, 7, 8, 10\right}$

Find the cardinality of $A\cup A$ .

Explanation

The question is asking us to find the number of elements in the union of A and A .

We simply combine the two sets together, and count the number of elements. If there are any elements that are the same, we keep one of them, and not both.

$A\cup A = \left{9, 10, 16, 17, 19\right}$

Now we simply count the number of elements in this set.

$\A= \left{12, 22, 26, 28, 29\right} \B= \left{1, 4, 14, 15, 22, 24\right} \C= \left{\varnothing, 2, 4, 8, 13, 14\right}$

Find the cardinality of $A\cap A$ .

Explanation

The question is asking us to find the number of elements in the intersection of A and A .

We simply find the common elements of the two sets, and count the number of elements.

$A\cap A = \left{12, 22, 26, 28, 29\right}$

Now we simply count the number of elements in this set.

$\A= \left{3, 8, 10, 21, 27, 29\right} \B= \left{5, 10, 11, 18, 21\right} \C= \left{\varnothing, 1, 2, 7, 9\right}$

Find the cardinality of $B\cap B$ .

Explanation

The question is asking us to find the number of elements in the intersection of B and B .

We simply find the common elements of the two sets, and count the number of elements.

$B\cap B = \left{5, 10, 11, 18, 21\right}$

Now we simply count the number of elements in this set.

$\A= \left{1, 2, 4, 6, 11, 21\right} \B= \left{3, 10, 13, 14, 21, 22\right} \C= \left{2, 4, 5, 8, 12, 14\right}$

Find the cardinality of $A\cap B$ .

Explanation

The question is asking us to find the number of elements in the intersection of A and B .

We simply find the common elements of the two sets, and count the number of elements.

$A\cap B = \left{21\right}$

Now we simply count the number of elements in this set.

$\A= \left{5, 10, 12, 13, 18, 21\right} \B= \left{\varnothing, 3, 4, 7, 10, 16\right} \C= \left{2, 4, 5, 7, 13\right}$

Find the cardinality of $A\cap B$ .

Explanation

The question is asking us to find the number of elements in the intersection of A and B .

We simply find the common elements of the two sets, and count the number of elements.

$A\cap B = \left{10\right}$

Now we simply count the number of elements in this set.

$\A= \left{6, 7, 10, 11, 19, 24\right} \B= \left{5, 8, 15, 17, 19, 28\right} \C= \left{1, 2, 3, 7, 9, 14\right}$

Find the cardinality of $A\cap B$ .

Explanation

The question is asking us to find the number of elements in the intersection of A and B .

We simply find the common elements of the two sets, and count the number of elements.

$A\cap B = \left{19\right}$
Now we simply count the number of elements in this set.

$\A= \left{11, 12, 13, 14, 18\right} \B= \left{1, 10, 11, 14, 16\right} \C= \left{3, 6, 8, 10, 17\right}$

Find the cardinality of $B\cup A$ .

Explanation

The question is asking us to find the number of elements in the union of B and A .

We simply combine the two sets together, and count the number of elements. If there are any elements that are the same, we keep one of them, and not both.

$B\cup A = \left{1, 10, 11, 12, 13, 14, 16, 18\right}$

Now we simply count the number of elements in this set.

Functions, Relations, & Cardinality

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