# Sum and subtraction with sets and matrix with formal notation

Hello I have some problems with formal notations, I have to sum (do the union?) of two sets and the sum (union?) of a set and a row of a matrix, is this notation correct?

1) Imagine $T_1 = \{1,2\}$ and $T_2 = \{2,3\}$, will $$\Sigma^{2}_{i=1}T_i$$ give me $\{1, 2, 3\}$ ? or should I use a different notation?

2) Imagine an adjacency matrix $n · n$ called $A$ and consider the first row $[1,j]A$ as [0,0,1,0,2] and I want to transform it in a set and add the elements in a set of the previous T, for example imagine $T_1=\{1,4\}$, will

$$T_1 + [1,j]A$$ give me $\{0,1,2,4\}$ ? or should I use a different notation?

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1) No, that doesn't work. I'm not sure what $x$ is, but the output of $\sum$ is a number, not a set. What you want to use instead is the union, where a large union can be denoted by

$$U_1 = \bigcup_{i=1}^2 \ T_i = T_1 \cup T_2 = \{1,2\} \cup \{2,3\} = \{1,2,3\}$$

2) Again, use the cup. If $A(1,\cdot)$ is the set of the elements of the first row of $A$, then

$$U_2 = A(1,\cdot) \cup T_1 = \{0,0,1,0,2\} \cup \{1,4\} = \{0,1,2,4\}$$

Representing a row of a matrix as a set is not so common though. Better may be

$$U_2 = \bigcup_{j=1}^n \{A_{1,j}\} \cup T_1 = \{0,0,1,0,2\} \cup \{1,4\} = \{0,1,2,4\}$$

where $A_{i,j}$ is the entry in the matrix $A$ at position $i,j$.

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how do I do minus in sets? – graphtheory92 Sep 12 '11 at 12:50
@Nicola: The setwise minus is written as $A \setminus B = \{x \in A: x \notin B\}$ (i.e. consists of elements in $A$ but not in $B$). Finally the intersection is written as a cap: $A \cap B = \{x \in A: x \in B\}$ (i.e. consists only of elements both in $A$ and $B$), and for the intersection you also have a bigger sign for large intersections: $\bigcap_{i=1}^n A_i = A_1 \cap A_2 \cap \ldots \cap A_n$. – TMM Sep 12 '11 at 13:59