B∪ C= 3,4,5,6,7,8 A∪ (B ∪ C)= 1,2,3,4,5,6,7,8 (i) A∪ B= 1,2,3,4,5,6 (A∪ B)∪ C= 1,2,3,4,5,6,7,8 (ii) From ...

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Associative Property

A ∪ B ∪ C= A ...

Question

$A \cup (B \cup C) = (A \cup B) \cup C$

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Solution

$B \cup C =$ { $3, 4, 5, 6, 7, 8$ }
$A \cup (B \cup C) =$ { $1, 2, 3, 4, 5, 6, 7, 8$ } $- (i)$

$A \cup B =$ { $1, 2, 3, 4, 5, 6$ }
$(A \cup B) \cup C =$ { $1, 2, 3, 4, 5, 6, 7, 8$ } $- (i i)$

From $(i)$ and $(i i)$
$A \cup (B \cup C) = (A \cup B) \cup C$

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|\begin{matrix} a + b & b + c & c + a \\ b + c & c + a & a + b \\ c + a & a + b & b + c \end{matrix}| = 2 |\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}|

|\begin{matrix} a & b - c & c - b \\ a - c & b & c - a \\ a - b & b - a & c \end{matrix}| = (a + b - c) (b + c - a) (c + a - b)

|\begin{matrix} a + b + c & - c & - b \\ - c & a + b + c & - a \\ - b & - a & a + b + c \end{matrix}| = 2 (a + b) (b + c) (c + a)

|\begin{matrix} - b c & b^{2} + b c & c^{2} + b c \\ a^{2} + a c & - a c & c^{2} + a c \\ a^{2} + a b & b^{2} + a b & - a b \end{matrix}| = {(a b + b c + c a)}^{3}

Verify the cyclic symmetry of the following expressions if a, b and c are the variables:

(1) a(b − c) + b(c − a) + c(a − b)

(2) ab(a + b) + bc(b + c) + ca(c + a)

(3) a²b(a − b) + b²c(b − c) + c²a(c − a)

(4) a²(a + b) + b²(b + c) + c²(c + a)

(5) ab²(a − b) + bc²(b − c) + ca²(c − a)

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