Własności działań
1. Czy podane działanie posiada element neutralny?
(a) ∗ : Z × Z → Z, a ∗ b = a + b (b) ∗ : Z × Z → Z, a ∗ b = ab
(c) ∗ : Q \ {0} × Q \ {0} → Q \ {0}, a ∗ b = a/b (d) ∗ : Z × Z → Z, a ∗ b = a + b
(e) ∗ : R × R → R, a ∗ b = a − b
(f) ∗ : {0, 1} × {0, 1} → {0, 1}, 0 ∗ 1 = 0, 1 ∗ 0 =
0, 1 ∗ 1 = 0, 0 ∗ 0 = 0
(g) ∗ : {0, 1} × {0, 1} → {0, 1}, 0 ∗ 1 = 0, 1 ∗ 0 = 0, 1 ∗ 1 = 1, 0 ∗ 0 = 0
(h) ∗ : R × R → R, a ∗ b = a + b + 8
(i) ∗ : R \ {0} × R \ {0} → R \ {0}, a ∗ b = a/b (j) ∗ : R+× R+→ R+, a ∗ b = ab
2. Dla podanych działań z elementem neutralnym, sprawdź, czy posiada ono element odwrotny do x. Jesli tak, to go wyznacz.
(a) ∗ : Z × Z → Z, a ∗ b = ab, x = 5
(b) ∗ : Z5\ {0} × Z5\ {0} → Z5\ {0}, a ∗ b = ab(
mod 5), x = 4
(c) ∗ : Q × Q → Q, a ∗ b = ab, x = 5 (d) ∗ : R × R → R, a ∗ b = ab, x = 4
(e) ∗ : Z × Z → Z, a ∗ b = ab, x = 1
(f) ∗ : Z × Z → Z, a ∗ b = a + b, x = 80 (g) ∗ : R × R → R, a ∗ b = ab + a + b, x = 3 (h) ∗ : {1} × {1} → {1}, 1 ∗ 1 = 1, x = 1
(i) ∗ : {1, 2, 3} × {1, 2, 3} → {1, 2, 3}, 1 ∗ a = a, a ∗ 1 = a, 2 ∗ 3 = 2, 3 ∗ 2 = 2, x = 3
(j) ∗ : R × R → R, a ∗ b = a + b, x = 15
3. Jeśli istnieją, wyznacz elementy odwrtone do podanych.
(a) ∗ : Z × Z → Z, a ∗ b = ab − 9, x = 9 (b) ∗ : Z × Z → Z, a ∗ b = a + b − 9, x = 9
(c) ∗ : Z × Z → Z, a ∗ b = a2+ b2, x = 9
(d) ∗ : Q \ {0} × Q \ {0} → Q \ {0}, a ∗ b = a/b, x = 2 (e) ∗ : Z4\ {0} × Z4\ {0} → Z4\ {0}, a ∗ b = ab(
mod 4), x = 2
(f) ∗ : Z6\ {0} × Z6\ {0} → Z6\ {0}, a ∗ b = ab(
mod 6), x = 5
(g) ∗ : Z5\ {0} × Z5\ {0} → Z5\ {0}, a ∗ b = a + b(
mod 5), x = 4
(h) ∗ : Zn\ {0} × Zn\ {0} → Zn\ {0}, a ∗ b = ab(
mod n), x = a
(i) ∗ : R × R → R, a ∗ b = a − b, x = 7 (j) ∗ : R × R → R, a ∗ b = a + b, x = 12
4. Czy podane działanie jest łączne?
(a) ∗ : Z × Z → Z, a ∗ b = ab + a + b (b) ∗ : R × R → R, a ∗ b = ab
(c) ∗ : R × R → R, a ∗ b = a + b (d) ∗ : R × R → R, a ∗ b = a − b (e) ∗ : R+× R+→ R+, a ∗ b = ab
(f) ∗ : R \ {0} × R \ {0} → R \ {0}, a ∗ b = a/b (g) składanie funkcji na zbiorze funkcji z R w R (h) ∗ : R × R → R, a ∗ b = 1
(i) ∗ : R × R → R, a ∗ b = c (j) ∗ : R × R → R, a ∗ b = b
5. Czy podane działanie jest przemienne?
(a) ∗ : R × R → R, a ∗ b = a + b (b) ∗ : R × R → R, a ∗ b = a − b
(c) ∗ : R × R → R, a ∗ b = a2+ b3 (d) ∗ : Z × Z → Z, a ∗ b = (−1)ab
(e) ∗ : Z × Z → Z, a ∗ b = (−1)a+b (f) ∗ : R × R → R, a ∗ b = min{a, b}
(g) ∗ : R × R → R, a ∗ b = max{a, b}
(h) ∗ : Q \ {0} × Q \ {0} → Q \ {0}, a ∗ b = a/b (i) ∗ : {1, 2} × {1, 2} → {1, 2}, 1 ∗ 2 = 1, 2 ∗ 1 =
1, 1 ∗ 1 = 1, 2 ∗ 2 = 2
(j) ∗ : R+× R+→ R+, a ∗ b = ba
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