For arbitrary subspaces U, V and W of a finite dimensional vector space, Which of the following hold: (a) U∩(V W) C U∩V U∩W (b) U∩(V W) ⊃U∩V U∩W. (c) (U∩V) WC (U W)∩ (V W). (d) (U∩V) W ⊃(U W)∩ (V W).?

Question

Answer

Proof for each option:

(a) U∩(V ∩ W) ⊆ U∩V ∩ U∩W:
Let x be an arbitrary element in U∩(V ∩ W). This means that x belongs to both U and (V ∩ W). Therefore, x belongs to U and x belongs to V ∩ W. From this, we can conclude that x belongs to U and x belongs to V and x belongs to W. Hence, x belongs to U∩V and x belongs to U∩W. Therefore, x belongs to U∩V ∩ U∩W. Since x was arbitrary, we have shown that U∩(V ∩ W) ⊆ U∩V ∩ U∩W.

(b) U∩(V ∩ W) ⊇ U∩V ∩ U∩W:
Let x be an arbitrary element in U∩V ∩ U∩W. This means that x belongs to both U∩V and U∩W. Therefore, x belongs to U and x belongs to V, and x belongs to U and x belongs to W. From this, we can conclude that x belongs to both U and (V ∩ W). Hence, x belongs to U∩(V ∩ W). Therefore, x belongs to U∩(V ∩ W). Since x was arbitrary, we have shown that U∩(V ∩ W) ⊇ U∩V ∩ U∩W.

(c) (U∩V) ∩ (U∩W) ⊆ (U∩W)∩(V∩W):
Let x be an arbitrary element in (U∩V) ∩ (U∩W). This means that x belongs to both U∩V and U∩W. Therefore, x belongs to U and x belongs to V, and x belongs to U and x belongs to W. From this, we can conclude that x belongs to (U∩W) and x belongs to (V∩W). Hence, x belongs to (U∩W)∩(V∩W). Therefore, x belongs to (U∩W)∩(V∩W). Since x was arbitrary, we have shown that (U∩V) ∩ (U∩W) ⊆ (U∩W)∩(V∩W).

(d) (U∩V) ∩ W ⊆ (U∩W) ∩ (V∩W):
Let x be an arbitrary element in (U∩V) ∩ W. This means that x belongs to both U∩V and W. Therefore, x belongs to U and x belongs to V, and x belongs to W. From this, we can conclude that x belongs to (U∩W) and x belongs to (V∩W). Hence, x belongs to (U∩W) ∩ (V∩W). Therefore, x belongs to (U∩W) ∩ (V∩W). Since x was arbitrary, we have shown that (U∩V) ∩ W ⊆

Answer

I think obtion b is correct

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