Union and Intersection

You can find the union of two or more finite sets with Evaluate, by using the symbol ∪ between the sets.

$\blacktriangleright$ Evaluate

$\left\{\vphantom{ 1,2,3}\right.$1, 2, 3$\left.\vphantom{ 1,2,3}\right\}$∪$\left\{\vphantom{ a,b,c}\right.$a, b, c$\left.\vphantom{ a,b,c}\right\}$ =  6pt$\left\{\vphantom{ 1,2,3,a,b,c}\right.$1, 2, 3, a, b, c$\left.\vphantom{ 1,2,3,a,b,c}\right\}$

$\left\{\vphantom{ 1,2,3}\right.$1, 2, 3$\left.\vphantom{ 1,2,3}\right\}$∪$\left\{\vphantom{ 3,5}\right.$3, 5$\left.\vphantom{ 3,5}\right\}$∪$\left\{\vphantom{ 7}\right.$7$\left.\vphantom{ 7}\right\}$ 6pt = $\left\{\vphantom{ 1,2,3,5,7}\right.$1, 2, 3, 5, 7$\left.\vphantom{ 1,2,3,5,7}\right\}$

$\left\{\vphantom{ \sqrt{2},\pi ,3.9,r}\right.$$\sqrt{{2}}$, π, 3.9, r$\left.\vphantom{ \sqrt{2},\pi ,3.9,r}\right\}$∪$\left\{\vphantom{ a,b,c}\right.$a, b, c$\left.\vphantom{ a,b,c}\right\}$ = $\left\{\vphantom{ \pi ,r,a,b,c,3.\,9,\sqrt{2}}\right.$π, r, a, b, c, 3. 9,$\sqrt{{2}}$$\left.\vphantom{ \pi ,r,a,b,c,3.\,9,\sqrt{2}}\right\}$

You can find the intersection of two or more finite sets with Evaluate, using the symbol ∩ between the sets.

$\blacktriangleright$ Evaluate

$\left\{\vphantom{ 1,2,3}\right.$1, 2, 3$\left.\vphantom{ 1,2,3}\right\}$∩$\left\{\vphantom{ 2,4,6}\right.$2, 4, 6$\left.\vphantom{ 2,4,6}\right\}$ = $\left\{\vphantom{ 2}\right.$2$\left.\vphantom{ 2}\right\}$ 6pt

$\left\{\vphantom{ a,b,c,d}\right.$a, b, c, d$\left.\vphantom{ a,b,c,d}\right\}$∩$\left\{\vphantom{ d,e,f}\right.$d, e, f$\left.\vphantom{ d,e,f}\right\}$ = $\left\{\vphantom{ d}\right.$d$\left.\vphantom{ d}\right\}$ 6pt

$\left\{\vphantom{ 1,2,3}\right.$1, 2, 3$\left.\vphantom{ 1,2,3}\right\}$∩$\left\{\vphantom{ a,b,c}\right.$a, b, c$\left.\vphantom{ a,b,c}\right\}$ = ∅ 6pt

$\left\{\vphantom{ 1,2,3}\right.$1, 2, 3$\left.\vphantom{ 1,2,3}\right\}$∩$\left\{\vphantom{ {}}\right.$$\left.\vphantom{ {}}\right\}$ = ∅

If two sets have no elements in common their intersection is the empty set, denoted by empty brackets $\left\{\vphantom{ {}}\right.$$\left.\vphantom{ {}}\right\}$ or the symbol ∅. To enter symbol for the empty set, click itbpF0.3105in0.2888in0.0701inmiscellaneous.wmf and select it from the Miscellaneous Symbols panel that appears.

You can evaluate combinations of union and intersection.

$\blacktriangleright$ Evaluate

$\left\{\vphantom{ 1,2,3,c}\right.$1, 2, 3, c$\left.\vphantom{ 1,2,3,c}\right\}$∩$\left(\vphantom{ \left\{ 2,4,6\right\} \cup \left\{ a,b,c\right\} }\right.$$\left\{\vphantom{ 2,4,6}\right.$2, 4, 6$\left.\vphantom{ 2,4,6}\right\}$∪$\left\{\vphantom{ a,b,c}\right.$a, b, c$\left.\vphantom{ a,b,c}\right\}$$\left.\vphantom{ \left\{ 2,4,6\right\} \cup \left\{ a,b,c\right\} }\right)$ = $\left\{\vphantom{ 2,c}\right.$2, c$\left.\vphantom{ 2,c}\right\}$ 6pt

$\left(\vphantom{ \left\{ 1,2,3,c\right\} \cap \left\{ 2,4,6\right\} }\right.$$\left\{\vphantom{ 1,2,3,c}\right.$1, 2, 3, c$\left.\vphantom{ 1,2,3,c}\right\}$∩$\left\{\vphantom{ 2,4,6}\right.$2, 4, 6$\left.\vphantom{ 2,4,6}\right\}$$\left.\vphantom{ \left\{ 1,2,3,c\right\} \cap \left\{ 2,4,6\right\} }\right)$∪$\left(\vphantom{ \left\{ 1,2,3,c\right\} \cap \left\{ a,b,c\right\} }\right.$$\left\{\vphantom{ 1,2,3,c}\right.$1, 2, 3, c$\left.\vphantom{ 1,2,3,c}\right\}$∩$\left\{\vphantom{ a,b,c}\right.$a, b, c$\left.\vphantom{ a,b,c}\right\}$$\left.\vphantom{ \left\{ 1,2,3,c\right\} \cap \left\{ a,b,c\right\} }\right)$ = $\left\{\vphantom{ 2,c}\right.$2, c$\left.\vphantom{ 2,c}\right\}$