Subgroup

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This interactive quiz is designed to test your knowledge on Subgroups. Answer this using multiple-choice questions to improve your understanding in a fun and engaging way.

Créé par

Ycel Marie Cruz

Philippines

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Subgroup


SubgroupVersion en ligne This interactive quiz is designed to test your knowledge on Subgroups. Answer this using multiple-choice questions to improve your understanding in a fun and engaging way. par Ycel Marie Cruz 1 Is (2Z, +) a subgroup of (Z, +)? Using theorem 2, what would be the next step? Choose one or more answers a (i) (2Z, +) is nonempty. b (ii) (2Z, +) is closed. c iii). Each element of (2Z, +) has an inverse element. 2 Since Z = (..., -2, 1, 0, 1, 2) then 2Z = ? a (..., -4, -2, 0, 1, 2, 4) . Clearly, (2Z, +) is nonempty b (..., -4, -2, -1, 0, 2, 4) . Clearly, (2Z, +) is nonempty c (..., -4, -2, 0, 2, 4) . Clearly, (2Z, +) is nonempty 3 Using theorem 2, what would be the next step? a (i) (2Z, +) is nonempty. b (ii) (2Z, +) is closed. c iii). Each element of (2Z, +) has an inverse element. 4 WST: a + b ∈ 2Z. By addition a + b = 2m + 2n. What would be the next step? a Factor out 2 a + b = 2 (m+n) b Factor out m a + b = m (2+n) c Tranpose the m+n a + b + m + n = 2 5 Since m, n ∈ Z then m+n ∈ Z and thus? a a + b ∈ Z Clearly (2Z, +) is closed. b m + n ∈ Z Clearly, (2Z, +) is closed. c a + b + m + n ∈ Z Clearly, (2Z, +) is closed. 6 Each element of (2Z, +) has an inverse element. Let: a=2m where m ∈ Z clearly a ∈ 2Z What is WST? a WST: a ¹ ∈ Z b WST: a ¹ ∈ 2Z c WST: a -¹ ∈ 2Z 7 What is the next step after WST? a Using the concept of inverse aa -¹ = e b Using the concept of inverse identity aa -¹ = e c Using the concept of identity a*a -¹ = e 8 Since the operation is in addition clearly e = 0. By replacement _________? a 2m + a ¹ = 0 b 2m + a -¹ = 0 c m + a -¹ = 0 9 Logically, for the equation to be true a-¹ = -2m Factor out 2 a-¹ = 2 (-m) What is the next step? a Since m ∈ Z thus -m ∈ Z . Evidently a-¹ ∈ 2Z b Since m ∈ Z thus -m ∈ Z . Evidently a¹ ∈ 2Z c Since m ∈ Z thus -m ∈ Z . Evidently a-¹ ∈ Z 10 Thus each element (2Z, +) has an? a inverse identity element b identity element c inverse element 11 Since all three properties were satisfied (2Z, +) a subgroup of? a (Z, +) b (2Z, +) c (2Z)

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Subgroup

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À propos de cette activité

Créé par

Téléchargez la version pour jouer sur papier

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SubgroupVersion en ligne

par Ycel Marie Cruz

Is (2Z, +) a subgroup of (Z, +)? Using theorem 2, what would be the next step?

Choose one or more answers

Since Z = (..., -2, 1, 0, 1, 2) then 2Z = ?

Using theorem 2, what would be the next step?

WST: a + b ∈ 2Z. By addition a + b = 2m + 2n. What would be the next step?

Since m, n ∈ Z then m+n ∈ Z and thus?

Each element of (2Z, +) has an inverse element. Let: a=2m where m ∈ Z clearly a ∈ 2Z What is WST?

What is the next step after WST?

Since the operation is in addition clearly e = 0. By replacement _________?

Logically, for the equation to be true a-¹ = -2m Factor out 2 a-¹ = 2 (-m) What is the next step?

Thus each element (2Z, +) has an?

Since all three properties were satisfied (2Z, +) a subgroup of?