Introducing Math Symbols for Union and Intersection

Our reader’s question is: “There are two sets of symbols for ‘union’ and ‘intersection’. One is the ∪ and the upside down ∩, and the other set is the ∨ and the ∧. What is the relationship between these symbols, which we sometimes think of as ‘U’s and ‘V’s?”

So let’s mind our Us and Vs, and their upside-down companions, in unions and intersections as well as logical functions.

The Math Symbols from the Question about Union and Intersection

The “V” symbols in the reader’s question are ∨ and ∧, which mean “Logical Or” and “Logical And.” The ∧ is a capital Greek Lambda.

The small ^ or “caret” is available on most keyboards as “shift-6”; it symbolizes the exponentiation function. It is important not to confuse ^ with ∧.

The symbol for “Union of sets” is ‘∪‘, while the symbol for “intersection of sets” is ‘∩.’

Set Theory for Union and Intersection

The approach which relates most closely to the question involves set theory. Let A={a, e, i, o, u, y} and B={a, b, c, d, e, f}.

The union of sets ‘A’ and ‘B’ is the set containing the unique elements found in either set ‘A’ or set ‘B’, or both. In other words, “Bring all the elements together but discard duplicates”. A∪B={a, b, c, d, e, f, i, o, u, y}.

The intersection of sets ‘A’ and ‘B’ is the set containing the unique elements from both set ‘A’ and set ‘B’. In other words, to create an intersection, only select elements found in both original sets, which are the duplicates discarded by the union operation. A∩B={a, e}

With reference to the original question, from the view of set theory, the word “union” relates to the ‘∪‘ symbol; and the word “intersection” relates to the ‘∩’ symbol.

 

Our reader’s question also asked about ‘V’ and upside-down ‘V’, ‘Λ’ or Lambda. These are used in mathematical logic.

Let statement A = “All humans are mammals.” Let B = “All mammals are humans.” Let C = “Some birds can fly under some conditions.” Both ‘A’ and ‘C’ are true statements, but ‘B’ is false.

Let ‘X’ and ‘Y’ represent any possibly true or false statements.

In the math of logic, the statement “X and Y”, or “XΛY”, is true if and only if both ‘X’ and ‘Y’ are true.

However, “X or Y”, or “X∨Y”, is false if and only if both ‘X’ and ‘Y’ are false. “X∨Y” is true if either ‘X’ or ‘Y’ is true, which includes the situation where both ‘X’ and ‘Y’ are true.

From the example statements, “A∨B”, “A∨C” and “B∨C” are all true. However, “AΛB” and “BΛC” are both false. Only “AΛC” is true because each of statements ‘A’ and ‘C’ are true.

Programming Computers with “And” and “Or”

Depending on the programming language, “X and Y” might be represented as “X&&Y” rather than “XΛY.” Likewise, “X or Y” might be shown as “X||Y” rather than “X∨Y.”

Exponentiation with the Caret Symbol

The caret symbol, ‘^’, might be mistaken for lambda, ‘Λ’. However, it usually represents the exponentiation operation. For example, 2^3 = 2*2*2 = 8.

When writing on paper, or when the text editor supports superscript, the exponent is shown as a superscript. See the image above.

Summary of Mathematical Symbols for Intersection and Union, And and Or

In set theory, intersection and union are shown by ‘∩’ and ‘∪‘. In mathematical logic, the “and” and “or” operations are shown by ‘Λ’ and ‘V’.

The union of sets, “A∪B”, might be seen as taking all the elements of ‘A’ and also the elements of ‘B’; but that would not be the “and” (‘Λ’) of mathematical logic.

The intersection of sets, “A∩B” has even less to do with a logical “or” (‘∨‘) operation.

Other areas of math may use these symbols in other ways, but these interpretations deal most directly with the reader’s question.

References:

Wolfram Mathematica Documentation. Intersection; Union; And; Or. (2012). Accessed July 26, 2012.

Wood, Alan. Symbol font – Unicode alternatives for Greek and special characters in HTML. (1997-2010). Accessed July 26, 2012.